# Proceedings of the Twenty-Sixth ICMI Study: Advances in Geometry Education

Angel Gutierrez Rodrigues, Thomas Lowrie, Fabien Emprin, Maha Abboud, Jonathan Adams, Jill Adler, Wilfredo Alangui, Angel Gutierrez, Alberto Arnal-Bailera, Francine Athias, et al.

<figure><img src="image_1.png" /></figure>

## To cite this version:

Angel Gutierrez Rodrigues, Thomas Lowrie, Fabien Emprin, Maha Abboud, Jonathan Adams, et al. "Proceedings of the Twenty-Sixth ICMI Study: Advances in Geometry Education." 26th ICMI study: Advances in Geometry Education, IREM de Reims, 2024, 978-2-910076-17-7. hal-04577863v1

HAL Id: hal-04577863

https://hal.science/hal-04577863v1

Submitted on 16 May 2024 (v1), last revised 21 Jun 2024 (v2)

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
# Proceedings of the Twenty-Sixth ICMI Study: Advances in Geometry Education

Angel Gutierrez Rodrigues, Thomas Lowrie, Fabien Emprin

► To cite this version:

Angel Gutierrez Rodrigues, Thomas Lowrie, Fabien Emprin. Proceedings of the Twenty-Sixth ICMI Study: Advances in Geometry Education. 26th ICMI study: Advances in Geometry Education, IREM de Reims, 2024, 978-2-910076-17-7. hal-04577863v2

HAL Id: hal-04577863

https://hal.science/hal-04577863v2

Submitted on 26 Sep 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Proceedings of the
Twenty-Sixth ICMI Study
Advances in Geometry Education

<figure><img src="image_3.png" /></figure>

Editors:

Thomas Lowrie

Angel Gutiérrez

Fabien Emprin

Reims, France

INSPÉ de l'Académie de Reims

Université de Reims Champagne-Ardenne

April 23 – 26, 2024
Proceedings of the
Twenty-Sixth ICMI Study
Advances in Geometry Education

<figure><img src="image_2.png" /></figure>

Editors:

Thomas Lowrie

Angel Gutiérrez

Fabien Emprin

Reims, France

INSPÉ de l'Académie de Reims

Université de Reims Champagne-Ardenne

April 23 – 26, 2024
<figure><img src="image_4.png" /></figure>

The Twenty-Sixth ICMI Study
Advances in Geometry Education

CONFERENCE PROCEEDINGS

EDITORS:

Thomas Lowrie

Angel Gutiérrez

Fabien Emprin

University of Canberra

Universidad de Valencia

Université de Reims

Reims, France

INSPÉ. Université de Reims Champagne-Ardenne

April 23 – 26, 2024

<figure><img src="image_5.png" /></figure>
<figure><img src="image_3.png" /></figure>

The Twenty-Sixth ICMI Study
Advances in Geometry Education

CONFERENCE PROCEEDINGS

EDITORS:

Thomas Lowrie

Angel Gutiérrez

Fabien Emprin

University of Canberra

Universidad de Valencia

Université de Reims

Reims, France

INSPÉ. Université de Reims Champagne-Ardenne

April 23 – 26, 2024

<figure><img src="image_4.png" /></figure>
**International Program Committee**

<table><tr><td>Cathy Bruce</td><td>Trent University, Canada</td></tr><tr><td>Jean-Luc Dorier ***</td><td>University of Geneva, Switzerland</td></tr><tr><td>Fabien Emprin</td><td>University of Reims, France</td></tr><tr><td>Angel Gutiérrez *</td><td>University of Valencia, Spain</td></tr><tr><td>Keith Jones</td><td>Southampton University, United Kingdom</td></tr><tr><td>Roza Leikin</td><td>University of Haifa, Israel</td></tr><tr><td>Thomas Lowrie *</td><td>University of Canberra, Australia</td></tr><tr><td>Frederick K.S. Leung **</td><td>University of Hong Kong, Hong Kong SAR</td></tr><tr><td>Michela Maschietto</td><td>University of Modena e Reggio Emilia, Italy</td></tr><tr><td>Lisnet Mwadzaangati</td><td>University of Malawi, Malawi</td></tr><tr><td>Oi-Lam Ng</td><td>Chinese University of Hong Kong, Hong Kong SAR</td></tr><tr><td>Yukari Okamoto</td><td>University of California at Santa Barbara, USA</td></tr><tr><td>Milton Rosa</td><td>Federal University of Ouro Preto, Brazil</td></tr><tr><td>Manuel Santos-Trigo</td><td>CINVESTAV, National Polytechnic Institute, Mexico</td></tr></table>

* Co-chair

** ICMI President

*** ICMI General-Secretary

**Local Organizer Committee**

<table><tr><td>Alexandre Booms</td><td>INSPÉ; University of Reims (France)</td></tr><tr><td>Frederic Castel</td><td>INSPÉ; University of Reims (France)</td></tr><tr><td>Fabien Emprin</td><td>INSPÉ; University of Reims (France)</td></tr></table>
**International Program Committee**

Cathy Bruce

Jean-Luc Dorier ***

Fabien Emprin

Angel Gutiérrez *

Keith Jones

Roza Leikin

Thomas Lowrie *

Frederick K.S. Leung **

Michela Maschietto

Lisnet Mwadzaangati

Oi-Lam Ng

Yukari Okamoto

Milton Rosa

Manuel Santos-Trigo

Trent University, Canada

University of Geneva, Switzerland

University of Reims, France

University of Valencia, Spain

Southampton University, United Kingdom

University of Haifa, Israel

University of Canberra, Australia

University of Hong Kong, Hong Kong SAR

University of Modena e Reggio Emilia, Italy

University of Malawi, Malawi

Chinese University of Hong Kong, Hong Kong SAR

University of California at Santa Barbara, USA

Federal University of Ouro Preto, Brazil

CINVESTAV, National Polytechnic Institute, Mexico

* Co-chair

** ICMI President

*** ICMI General-Secretary

**Local Organizer Committee**

Alexandre Booms

Frederic Castel

Fabien Emprin

INSPÉ; University of Reims (France)

INSPÉ; University of Reims (France)

INSPÉ; University of Reims (France)
© 2024 left to the authors

All rights reserved

Éditeur/imprimeur: IREM de REIMS. 23 Rue Clément ADER, 51100 Reims (France)

ISBN: 978-2-910076-17-7

EAN: 9782910076177

Dépôt légal May 2024

Photo credits

Front cover: Argonautes' Tower, Reims, Ⓔ①②⑩ Fabien Emprin

Back cover: Champagne vineyard, Ⓔ①②⑩ Alexandre Booms
Bandstand of La Patte d'Oie Park, Reims, Ⓔ①②⑩ Fabien Emprin
Remains of the former Cordeliers Convent, Reims, Ⓔ①②⑩ Fabien Emprin
© 2024 left to the authors

All rights reserved

Éditeur/imprimeur: IREM de REIMS. 23 Rue Clément ADER, 51100 Reims (France)

ISBN: 978-2-910076-17-7

EAN: 9782910076177

Dépôt légal May 2024

Photo credits

Front cover: Argonautes' Tower, Reims, Ⓔ Ⓒ Ⓒ Ⓒ Ⓒ Fabien Emprin

Back cover: Champagne vineyard, Ⓔ Ⓒ Ⓒ Ⓒ Ⓒ Alexandre Booms
Bandstand of La Patte d'Oie Park, Reims, Ⓔ Ⓒ Ⓒ Ⓒ Ⓒ Fabien Emprin
Remains of the former Cordeliers Convent, Reims, Ⓔ Ⓒ Ⓒ Ⓒ Ⓒ Fabien Emprin
ACKNOWLEDGEMENTS

We are grateful to the following institutions and companies for their financial and material support to the IPC Meeting, held in Valencia (Spain) in February 2023, and the 26th ICMI Study Conference. Thanks to them, we were able to have this international conference with more than eighty participants from thirty-five countries.

<table><tr><td>irem</td><td>Assemblée des Directeurs d'IREM - Réseau National des IREM (ADIREM) (France)</td></tr><tr><td></td><td>Association pour la Recherche en Didactique des Mathématiques (ARDM) de Reims (France)</td></tr><tr><td>IREM Reims</td><td>Institut de Recherche sur l'Enseignement des Mathématiques (IREM) de Reims (France)</td></tr><tr><td>INSPE Institut national supérieur du professorat et de l'éducation Académie de Reims</td><td>Institution National Supérieur de Professorat et de l'Éducation (INSPÉ) de l'Académie de Reims (France)</td></tr><tr><td>TRENT UNIVERSITY OFFICE OF RESEARCH & INNOVATION</td><td>Trent University (Canada)</td></tr><tr><td>UNIVERSITÉ DE REIMS CHAMPAGNE-ARDENNE</td><td>Université de Reims Champagne-Ardenne (France)</td></tr><tr><td>VNIVERSITAT VALÈNCIA</td><td>Universitat de València (Spain)</td></tr><tr><td>fepe</td><td>Federation of Parents' Councils of the Marne (France)</td></tr><tr><td>GO TRONIC ROBOTIQUE ET COMPOSANTS ÉLECTRONIQUES</td><td>Go Tronic has been distributing electronic components in France and abroad since 1990.</td></tr><tr><td>DAGOMA</td><td>Since 2013, DAGOMA has been designing technical and technological products.</td></tr><tr><td>NUMWORKS</td><td>The graphing calculator that makes everybody a math person.</td></tr><tr><td>TEXAS INSTRUMENTS</td><td>Our semiconductor chips can be found in every type of electronic system - from electric vehicles to industrial robots, solar panels to satellites.</td></tr></table>

We also express our deep appreciation to Lena Koch (IMU Secretariat) who assisted us in solving many administrative matters promptly and efficiently.
ACKNOWLEDGEMENTS

We are grateful to the following institutions and companies for their financial and material support to the IPC Meeting, held in Valencia (Spain) in February 2023, and the 26th ICMI Study Conference. Thanks to them, we were able to have this international conference with more than eighty participants from thirty-five countries.

<table><tr><td>irem</td><td>Assemblée des Directeurs d'IREM - Réseau National des IREM (ADIREM) (France)</td></tr><tr><td></td><td>Association pour la Recherche en Didactique des Mathématiques (ARDM) de Reims (France)</td></tr><tr><td>IREM Reims</td><td>Institut de Recherche sur l'Enseignement des Mathématiques (IREM) de Reims (France)</td></tr><tr><td>INSPF Institut national supérieur du professorat et de l'éducation Académie de Reims</td><td>Institution National Supérieur de Professorat et de l'Éducation (INSPÉ) de l'Académie de Reims (France)</td></tr><tr><td>TRENT UNIVERSITY</td><td>Trent University (Canada)</td></tr><tr><td>UNIVERSITÉ DE REIMS CHAMPAGNE-ARDENNE</td><td>Université de Reims Champagne-Ardenne (France)</td></tr><tr><td>VNIVERSITAT ID VALÈNCIA</td><td>Universitat de València (Spain)</td></tr><tr><td>FORGE</td><td>Federation of Parents' Councils of the Marne (France)</td></tr><tr><td>GO TRONIC ROBOTIQUE ET COMPOSÉRIS ELECTRONIQUES</td><td>Go Tronic has been distributing electronic components in France and abroad since 1990</td></tr><tr><td>DAGOMA</td><td>Since 2013, DAGOMA has been designing technical and technological products</td></tr><tr><td>NUMWORKS</td><td>The graphing calculator that makes everybody a math person</td></tr><tr><td>TEXAS INSTRUMENTS</td><td>Our semiconductor chips can be found in every type of electronic system - from electric vehicles to industrial robots, solar panels to satellites</td></tr></table>

We also express our deep appreciation to Lena Koch (IMU Secretariat) who assisted us in solving many administrative matters promptly and efficiently.
# INDEX

Preface xi

## Plenary lectures

Research on teaching knowledge in geometry: the case of proof in the United States
Herbst, P.
3

Teaching and learning geometry in early grades with technology
Sinclair, N.
5

## Plenary panel 1: Framing Geometric Representations and Practices in Culturally diverse Settings

Presentation
Lowrie, T.
7

Context and language in Malawai classrooms
Mwadzaangati, L.
11

Framing geometric representations and practices in culturally-diverse settings
Sgreccia, N.F.
15

A glimpse of the practices of geometry education in Iran
Gooya, Z.
21

Ethnomodels of Landless Peoples' Movement: Measuring Land - Brazil
Rosa, M.
25

## Plenary panel 2: Teaching geometry in France

Teaching geometry in France
Emprin, F., Audra, I., Binet, M., Da Motta, B., Foy, M.-P., Marche, A., Trouillet, C.
31

## Discussion Topics

### Topic A: Theoretical perspectives

Undergraduate mathematics students' reasoning and argumentation in problem-solving geometrical tasks
Chicalote-Jiménez, T.A., Ortiz-May, D.J., Gómez-Arciga, A.
37

Spatial reasoning in authentic contexts of an engineering challenge: tapping into children's intuitive understanding of relationships between objects and self
Freiman, V., Fellus, O.
45
# INDEX

Preface ix

## Plenary lectures

Research on teaching knowledge in geometry: the case of proof in the United States
Herbst, P.
3

Teaching and learning geometry in early grades with technology
Sinclair, N.
5

## Plenary panel 1: Framing Geometric Representations and Practices in Culturally diverse Settings

Presentation
Lowrie, T.
7

Context and language in Malawai classrooms
Mwadzaangati, L.
11

Framing geometric representations and practices in culturally-diverse settings
Sgreccia, N.F.
15

A glimpse of the practices of geometry education in Iran
Gooya, Z.
21

Ethnomodels of Landless Peoples' Movement: Measuring Land - Brazil
Rosa, M.
25

## Plenary panel 2: Teaching geometry in France

Teaching geometry in France
Emprin, F., Audra, I., Binet, M., Da Motta, B., Foy, M.-P., Marche, A., Trouillet, C.
31

## Discussion Topics

### Topic A: Theoretical perspectives

Undergraduate mathematics students' reasoning and argumentation in problem-solving geometrical tasks
Chicalote-Jiménez, T.A., Ortiz-May, D.J., Gómez-Arciga, A.
37

Spatial reasoning in authentic contexts of an engineering challenge: tapping into children's intuitive understanding of relationships between objects and self
Freiman, V., Fellus, O.
45
Identifying a sequence of core skills for deductive proving in secondary school geometry
Fujita, T., Kondo, Y., Kumakura, H., Miyawaki, S., Kunimune, S., Shojima, K., Jones, K.
53

The role of visual mediators in geometric learning processes in university education
Gambini, A., Viola, G., Ferretti, F.
61

From geometric thinking to geometric practice: the potential of representations of practice for teaching and learning geometry in secondary schools
Herbst, P., Chazan, D.
69

Proofs in geometry teaching in the Brazilian context: a representation of yesterday and today
Jahn, A.P., Leme da Silva, M.
77

A dynamic and spatial approach to enrich the teaching and learning of geometry in primary school
Marchand, P., Sinclair, N., Julien, N.
85

Geometric prediction as a bridging process between transforming and understanding
Miragliotta, E.
93

Building mathematical maturity through algebraic topology challenging paths
Miranda, A.
101

On the role of shifts of attention and figural apprehension in the evolution of geometric perception
Palatnik, A.
109

Spatial reasoning interventions and transfer to geometry: what we know about mechanism
Resnick, I., Adams, J.
117

Guided play that fosters the development of children's spatial ability
Watanabe, S.
125

## Topic B: Curricular and methodological approaches

Classroom simulators: a new training approach to investigate teachers' professional knowledge and support its development
Abboud, M., Emprin, F.
135

Expert judgment for content validation of a questionnaire on the level 5 definition process within the Van Hiele framework
Arnal-Bailera, A., Manero, V.
143
Identifying a sequence of core skills for deductive proving in secondary school geometry
Fujita, T., Kondo, Y., Kumakura, H., Miyawaki, S., Kunimune, S., Shojima, K., Jones, K.
53

The role of visual mediators in geometric learning processes in university education
Gambini, A., Viola, G., Ferretti, F.
61

From geometric thinking to geometric practice: the potential of representations of practice for teaching and learning geometry in secondary schools
Herbst, P., Chazan, D.
69

Proofs in geometry teaching in the Brazilian context: a representation of yesterday and today
Jahn, A.P., Leme da Silva, M.
77

A dynamic and spatial approach to enrich the teaching and learning of geometry in primary school
Marchand, P., Sinclair, N., Julien, N., Forest, M.
85

Geometric prediction as a bridging process between transforming and understanding
Miragliotta, E.
93

Building mathematical maturity through algebraic topology challenging paths
Miranda, A.
101

On the role of shifts of attention and figural apprehension in the evolution of geometric perception
Palatnik, A.
109

Spatial reasoning interventions and transfer to geometry: what we know about mechanism
Resnick, I., Adams, J.
117

Guided play that fosters the development of children's spatial ability
Watanabe, S.
125

## Topic B: Curricular and methodological approaches

Classroom simulators: a new training approach to investigate teachers' professional knowledge and support its development
Abboud, M., Emprin, F.
135

Expert judgment for content validation of a questionnaire on the level 5 definition process within the Van Hiele framework
Arnal-Bailera, A., Manero, V.
143
<table>
  <tbody>
    <tr>
      <td>Teaching and learning geometry of pre-service primary education teachers based on the visual-analytical method of directed observation<br>Baranovic, N.</td>
      <td>151</td>
    </tr>
    <tr>
      <td>Geometry teaching from Babylon to the computer era<br>Carvalho E Silva, J.</td>
      <td>159</td>
    </tr>
    <tr>
      <td>Definitions of prospective primary teachers concerning the area of 2D figures<br>Caviedes, S., De Gamboa, G., Badillo, E., Pino-Fan, L.</td>
      <td>167</td>
    </tr>
    <tr>
      <td>Problem types in geometry textbooks: Russia's experience<br>Karp, A.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>Casual and geometric praxeologies in a study of symmetric figures<br>Kondratieva, M.</td>
      <td>183</td>
    </tr>
    <tr>
      <td>Learning stakes targeted by teachers in a figure restoration activity<br>Mangiante-Orsola, C., Guille-Biel Winder, C.</td>
      <td>191</td>
    </tr>
    <tr>
      <td>What structure must the statement of a geometry task have to promote a certain type of argument?<br>Molina, O., Samper, C., Vargas, C., Camargo, L., Perry, P.</td>
      <td>199</td>
    </tr>
    <tr>
      <td>Analysis of visualization as an indicator of mathematical giftedness<br>Mora, M., Gutiérrez, A., Jaime, A.</td>
      <td>207</td>
    </tr>
    <tr>
      <td>Lesson study as a context for teachers' learning of language responsive geometry teaching<br>Mwadzaangati, L., Adler, J.</td>
      <td>215</td>
    </tr>
    <tr>
      <td>Multi-perspectivity: a 'red thread' through discussions on geometry for teaching and learning<br>Neubrand, M.</td>
      <td>223</td>
    </tr>
    <tr>
      <td>How to teach geometry in continuity along schooling?<br>Perrin-Glorian, M.-J., Mathé, A.-C., Celi, V., Bulf, C.</td>
      <td>231</td>
    </tr>
    <tr>
      <td>Approach to teaching geometry to dyspraxia students<br>Petitfour, E.</td>
      <td>239</td>
    </tr>
    <tr>
      <td>Language and mathematics: a case of geometric shape identification<br>Pieng, P., Okamoto, Y., Weckbacher, L.</td>
      <td>247</td>
    </tr>
    <tr>
      <td>Designing meaningful tasks to promote argumentation skills in DGE - a concept for a professional development program<br>Stroetmann, E., Kortenkamp, U.</td>
      <td>255</td>
    </tr>
    <tr>
      <td>The use of straightedge, compass, and fixed shapes to enhance the content knowledge of mathematics teachers in their professional training in geometry education<br>Yevdokimov, O.</td>
      <td>263</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>Teaching and learning geometry of pre-service primary education teachers based on the visual-analytical method of directed observation<br>Baranovic, N.</td>
      <td>151</td>
    </tr>
    <tr>
      <td>Geometry teaching from Babylon to the computer era<br>Carvalho E Silva, J.</td>
      <td>159</td>
    </tr>
    <tr>
      <td>Definitions of prospective primary teachers concerning the area of 2D figures<br>Caviedes, S., De Gamboa, G., Badillo, E., Pino-Fan, L.</td>
      <td>167</td>
    </tr>
    <tr>
      <td>Problem types in geometry textbooks: Russia's experience<br>Karp, A.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>Casual and geometric praxeologies in a study of symmetric figures<br>Kondratieva, M.</td>
      <td>183</td>
    </tr>
    <tr>
      <td>Learning stakes targeted by teachers in a figure restoration activity<br>Mangiante-Orsola, C., Guille-Biel Winder, C.</td>
      <td>191</td>
    </tr>
    <tr>
      <td>What structure must the statement of a geometry task have to promote a certain type of argument?<br>Molina, O., Samper, C., Vargas, C., Camargo, L., Perry, P.</td>
      <td>199</td>
    </tr>
    <tr>
      <td>Analysis of visualization as an indicator of mathematical giftedness<br>Mora, M., Gutiérrez, A., Jaime, A.</td>
      <td>207</td>
    </tr>
    <tr>
      <td>Lesson study as a context for teachers' learning of language responsive geometry teaching<br>Mwadzaangati, L., Adler, J.</td>
      <td>215</td>
    </tr>
    <tr>
      <td>Multi-perspectivity: a 'red thread' through discussions on geometry for teaching and learning<br>Neubrand, M.</td>
      <td>223</td>
    </tr>
    <tr>
      <td>How to teach geometry in continuity along schooling?<br>Perrin-Glorian, M.-J., Mathé, A.-C., Celi, V., Bulf, C.</td>
      <td>231</td>
    </tr>
    <tr>
      <td>Approach to teaching geometry to dyspraxia students<br>Petitfour, E.</td>
      <td>239</td>
    </tr>
    <tr>
      <td>Language and mathematics: a case of geometric shape identification<br>Pieng, P., Okamoto, Y., Weckbacher, L.</td>
      <td>247</td>
    </tr>
    <tr>
      <td>Designing meaningful tasks to promote argumentation skills in DGE - a concept for a professional development program<br>Stroetmann, E., Kortenkamp, U.</td>
      <td>255</td>
    </tr>
    <tr>
      <td>The use of straightedge, compass, and fixed shapes to enhance the content knowledge of mathematics teachers in their professional training in geometry education<br>Yevdokimov, O.</td>
      <td>263</td>
    </tr>
  </tbody>
</table>
## Topic C: Resources for teaching and learning geometry

Destructuring / restructuring of an artifact: the case of the number line
Aebischer, T. 273

Dynamic geometry in primary school
Athias, F. 281

The notion of angle and the GGBot as a tool-to-think-with ... or without
Baccaglini-Frank, A., Funghi, S., Miragliotta, E. 289

A structured practical activity to enhance understanding of circle circumference and π approximation
Bahramibidkalme, M., Gooya, Z., Gholamazad, S. 297

The development of sequential and discursive apprehension in kindergarten students when they build polygons
Bernabeu, M., Buforn, A., Castillo, S. 305

Knowledge and reasoning in circulation during a situation of figures reproduction by folding
Blanquart Henry, S., Guille-Biel Winder, C., Petitfour, E. 313

The digital interactive mathematical map for geometry
Brandl, M., Hackstein, U., Vinerean, M., Liljekvist, Y. 321

Grids as objects and tools for the geometry curriculum
Bruce, C., Sinclair, N., Bodnar, J., Jackiw, N. 329

Virtual environment for spatial knowledge, opportunities and learning objectives
Coutat, S., Dorier, J.-L. 337

Learning coordinate geometry with Scratch: task design from an embodied and APOS approach
Cui, Z., Ng, O.-L., Koo, C.M. 345

Contribution of gestures to the acquisition of geometric properties
Douaire, J., Emprin, F. 353

Connecting secondary and college geometry: resolution of problems of finding intersections and measures of curves using dynamic geometry software
Hoyos, V., Robles-Pecina, L. 361

How can virtual geometry manipulatives be used in ways that mitigate their ontological, technological and pedagogical limitations?
Kortenkamp, U., Larkin, K. 369

Supporting geometric reasoning for underserved students in India through connected learning initiative
Kumar, R., Srinivas, S. 377

Material and digital tools for geometry in mathematics laboratory
Maschietto, M. 385
## Topic C: Resources for teaching and learning geometry

Destructuring / restructuring of an artifact: the case of the number line
Aebischer, T. 273

Dynamic geometry in primary school
Athias, F. 281

The notion of angle and the GGBot as a tool-to-think-with ... or without
Baccaglini-Frank, A., Funghi, S., Miragliotta, E. 289

A structured practical activity to enhance understanding of circle circumference and $π$ approximation
Bahramibidkalme, M., Gooya, Z., Gholamazad, S. 297

The development of sequential and discursive apprehension in kindergarten students when they build polygons
Bernabeu, M., Buforn, A., Castillo, S. 305

Knowledge and reasoning in circulation during a situation of figures reproduction by folding
Blanquart Henry, S., Guille-Biel Winder, C., Petitfour, E. 313

The digital interactive mathematical map for geometry
Brandl, M., Hackstein, U., Vinerean, M., Liljekvist, Y. 321

Grids as objects and tools for the geometry curriculum
Bruce, C., Sinclair, N., Bodnar, J., Jackiw, N. 329

Virtual environment for spatial knowledge, opportunities and learning objectives
Coutat, S., Dorier, J.-L. 337

Learning coordinate geometry with Scratch: task design from an embodied and APOS approach
Cui, Z., Ng, O.-L., Koo, C.M. 345

Contribution of gestures to the acquisition of geometric properties
Douaire, J., Emprin, F. 353

Connecting secondary and college geometry: resolution of problems of finding intersections and measures of curves using dynamic geometry software
Hoyos, V., Robles-Pecina, L. 361

How can virtual geometry manipulatives be used in ways that mitigate their ontological, technological and pedagogical limitations?
Kortenkamp, U., Larkin, K. 369

Supporting geometric reasoning for underserved students in India through connected learning initiative
Kumar, R., Srinivas, S. 377

Material and digital tools for geometry in mathematics laboratory
Maschietto, M. 385
<table>
  <tbody>
    <tr>
      <td>Development of resources to study geometry reusing materials<br>Sgreccia, N., Schaefer, L., Grossi, S., Di Biaggio, B.</td>
      <td>393</td>
    </tr>
    <tr>
      <td>How do teachers coordinate students' empirical perception and logical reasoning in 3D dynamic geometry environments: Chinese and French cases<br>Shao, M.-Y., Trgalova, J., Trouche, L.</td>
      <td>401</td>
    </tr>
    <tr>
      <td>Analogies: a way to promote the learning of proof in 3D geometry using dynamic geometry environments<br>Sua, C., Gutiérrez, A., Jaime, A.</td>
      <td>409</td>
    </tr>
    <tr>
      <td>Challenges of integrating DGS with geometry education<br>Tavakoli, M., Gooya, Z.</td>
      <td>417</td>
    </tr>
    <tr>
      <td>Developing analytical thinking in the recognition of unusual geometrical shapes with young pupils (from 4-8-year-old)<br>Vendeira, C., Coutat, S.</td>
      <td>425</td>
    </tr>
    <tr>
      <td>Descriptive automated assessment: facilitating inquiry in geometry<br>Yerushamly, M., Olsher, S.</td>
      <td>433</td>
    </tr>
    <tr>
      <td colspan="2"><strong>Topic D: Multidisciplinary perspectives</strong></td>
    </tr>
    <tr>
      <td>Exploring creative problem-solving with eye-tracking methodology<br>Eraky, A., Hadad, B.-S., Hel-Or, H., Abboud, E., Leikin, R.</td>
      <td>443</td>
    </tr>
    <tr>
      <td>Teaching geometry to advance design justice<br>González, G., Shehab, S., Powers, E.</td>
      <td>451</td>
    </tr>
    <tr>
      <td>Visualization and spatial visualization in geometry<br>Harris, D., Logan, T., Lowrie, T.</td>
      <td>459</td>
    </tr>
    <tr>
      <td>Discovering geometry in African ethno artifacts<br>Milinkovic, J., Vorkapic, M.</td>
      <td>467</td>
    </tr>
    <tr>
      <td>Investigating geometric knowledge in the art of Sisal tapestry in a local community through ethnomodelling<br>Orey, D.C., Rosa, M., Rosa Filho, O.</td>
      <td>475</td>
    </tr>
    <tr>
      <td>A pedagogical action based on an ethnomathematical perspective for the development of geometric content for visually impaired students to improve the teaching practice of a visually impaired mathematics teacher<br>Rosa, M., Orey, D.C., Da Silva, G.A.P.</td>
      <td>483</td>
    </tr>
    <tr>
      <td colspan="2"><strong>The Discussion Document</strong></td>
    </tr>
    <tr>
      <td>The 26th ICMI Study: Advances in Geometry Education. Discussion Document<br>International Program Committee</td>
      <td>493</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>Development of resources to study geometry reusing materials<br>Sgreccia, N., Schaefer, L., Grossi, S., Di Biaggio, B.</td>
      <td>393</td>
    </tr>
    <tr>
      <td>How do teachers coordinate students' empirical perception and logical reasoning in 3D dynamic geometry environments: Chinese and French cases<br>Shao, M.-Y., Trgalova, J., Trouche, L.</td>
      <td>401</td>
    </tr>
    <tr>
      <td>Analogies: a way to promote the learning of proof in 3d geometry using dynamic geometry environments<br>Sua, C., Gutiérrez, A., Jaime, A.</td>
      <td>409</td>
    </tr>
    <tr>
      <td>Challenges of integrating DGS with geometry education<br>Tavakoli, M., Gooya, Z.</td>
      <td>417</td>
    </tr>
    <tr>
      <td>Developing analytical thinking in the recognition of unusual geometrical shapes with young pupils (from 4-8-year-old)<br>Vendeira, C., Coutat, S.</td>
      <td>425</td>
    </tr>
    <tr>
      <td>Descriptive automated assessment: facilitating inquiry in geometry<br>Yerushamly, M., Olsher, S.</td>
      <td>433</td>
    </tr>
    <tr>
      <td colspan="2"><strong>Topic D: Multidisciplinary perspectives</strong></td>
    </tr>
    <tr>
      <td>Exploring creative problem-solving with eye-tracking methodology<br>Eraky, A., Hadad, B.-S., Hel-Or, H., Abboud, E., Leikin, R.</td>
      <td>443</td>
    </tr>
    <tr>
      <td>Teaching geometry to advance design justice<br>González, G., Shehab, S., Powers, E.</td>
      <td>451</td>
    </tr>
    <tr>
      <td>Visualization and spatial visualization in geometry<br>Harris, D., Logan, T., Lowrie, T.</td>
      <td>459</td>
    </tr>
    <tr>
      <td>Discovering geometry in African ethno artifacts<br>Milinkovic, J., Vorkapic, M.</td>
      <td>467</td>
    </tr>
    <tr>
      <td>Investigating geometric knowledge in the art of Sisal tapestry in a local community through ethnomodelling<br>Orey, D.C., Rosa, M., Rosa Filho, O.</td>
      <td>475</td>
    </tr>
    <tr>
      <td>A pedagogical action based on an ethnomathematical perspective for the development of geometric content for visually impaired students to improve the teaching practice of a visually impaired mathematics teacher<br>Rosa, M., Orey, D.C., Da Silva, G.A.P.</td>
      <td>483</td>
    </tr>
    <tr>
      <td colspan="2"><strong>The Discussion Document</strong></td>
    </tr>
    <tr>
      <td>The 26th ICMI Study: Advances in Geometry Education. Discussion Document<br>International Program Committee</td>
      <td>493</td>
    </tr>
  </tbody>
</table>
# PREFACE

Our engagement with this ICMI Study Conference began when we accepted invitations from the ICMI President, Frederick Leung, to co-chair the *Study 26: Advances in Geometry Education*. Although we have both worked internationally in geometry education for decades, we had not previously met one another. A briefing meeting with the President, and the Secretary General, Jean-Luc Dorier, provided context and scope for the Study. As with all ICMI studies, the central aims are to consider the topic (in our case geometry education) with respect to theory, research, practice, and policy; and to suggest new directions of research that take into account contextual, cultural, national, and political dimensions of the field. We then worked collaboratively with Professors Leung, Dorier, and other members of the ICMI Executive, to establish an International Program Committee (IPC) with expertise in geometry education. The IPC members have invested substantial time and effort to be part of the IPC, over an extended period. We are grateful for their collective wisdom and technical expertise in framing the core work of the Study. We are especially appreciative of Professor Fabien Emprin's commitment to the Study by taking on the responsibility of Conference Convenor. Fabien's leadership and organisational skills became an integral part of the Study's success. We would also like to acknowledge the effort that some participants from economically disadvantaged countries have made to attend the conference, and the economic support we have received from ICMI and Trent University (Canada) to support their attendance.

The IPC met in Valencia in February 2023 to develop the ICMI Study 26 Discussion Document, included at the end of the Proceedings, which was finalized and distributed internationally in June 2023. We express our gratitude to the University of Valencia for providing working spaces and facilities during the three-day meeting. The Study Document served as a call for papers for the ICMI Study 26 Conference, hosted by the Université de Reims Champagne Ardenne, France, during April, 2024.

An ICMI Study Conference is quite different to a traditional conference. A majority of the time is devoted to the engagement of Working Groups organized around the topics and led by IPC members. During these Working Groups brief presentations by the participants, based on their papers, serve as a catalyst for in-depth exploration of the topics and sub-topics —with the primary purpose being the preparation and initial development of a published volume.

The activity of the Study is organized around four focused topics, aimed to provide complementary perspectives and approaches to the teaching and learning of geometry. Contributions to the topics were organized around sets of related sub-topics, each sub-topic focusing on a specific issue and stating a set of questions aimed to lead discussions, as defined in the Discussion Document. The four topics are:

A. Theoretical perspectives;

B. Curricular and methodological approaches;

C. Resources for teaching and learning geometry; and
# PREFACE

Our engagement with this ICMI Study Conference began when we accepted invitations from the ICMI President, Frederick Leung, to co-chair the *Study 26: Advances in Geometry Education*. Although we have both worked internationally in geometry education for decades, we had not previously met one another. A briefing meeting with the President, and the Secretary General, Jean-Luc Dorier, provided context and scope for the Study. As with all ICMI studies, the central aims are to consider the topic (in our case geometry education) with respect to theory, research, practice, and policy; and to suggest new directions of research that take into account contextual, cultural, national, and political dimensions of the field. We then worked collaboratively with Professors Leung, Dorier, and other members of the ICMI Executive, to establish an International Program Committee (IPC) with expertise in geometry education. The IPC members have invested substantial time and effort to be part of the IPC, over an extended period. We are grateful for their collective wisdom and technical expertise in framing the core work of the Study. We are especially appreciative of Professor Fabien Emprin's commitment to the Study by taking on the responsibility of Conference Convenor. Fabien's leadership and organisational skills became an integral part of the Study's success. We would also like to acknowledge the effort that some participants from economically disadvantaged countries have made to attend the conference, and the economic support we have received from ICMI and Trent University (Canada) to support their attendance.

The IPC met in Valencia in February 2023 to develop the ICMI Study 26 Discussion Document, included at the end of the Proceedings, which was finalized and distributed internationally in June 2023. We express our gratitude to the University of Valencia for providing working spaces and facilities during the three-day meeting. The Study Document served as a call for papers for the ICMI Study 26 Conference, hosted by the Université de Reims Champagne Ardenne, France, during April, 2024.

An ICMI Study Conference is quite different to a traditional conference. A majority of the time is devoted to the engagement of Working Groups organized around the topics and led by IPC members. During these Working Groups brief presentations by the participants, based on their papers, serve as a catalyst for in-depth exploration of the topics and sub-topics—with the primary purpose being the preparation and initial development of a published volume.

The activity of the Study is organized around four focused topics, aimed to provide complementary perspectives and approaches to the teaching and learning of geometry. Contributions to the topics were organized around sets of related sub-topics, each sub-topic focusing on a specific issue and stating a set of questions aimed to lead discussions, as defined in the Discussion Document. The four topics are:

A. Theoretical perspectives;

B. Curricular and methodological approaches;

C. Resources for teaching and learning geometry; and
D. Multidisciplinary perspectives.

These topics and related sub-topics provided the basis for inviting papers. Each submission was reviewed by at least two IPC members, and authors of accepted papers were invited to participate in the Study Conference. Accepted papers were then revised (when necessary) by the authors, before being published in these electronic conference proceedings. The conference proceedings include 60 papers from around the world. The countries represented in the Proceedings include: Argentina, Australia, Brazil, Canada, Chile, China, Colombia, Costa Rica, Croatia, France, Germany, Greece, Hong Kong, India, Iran, Israel, Italy, Japan, Malawi, Mexico, New Zealand, Portugal, Serbia, South Africa, Spain, Sweden, Switzerland, United Kingdom, and the United States.

In addition, we invited two esteemed scholars to present plenary lectures and another five to participate in a plenary panel on geometry practices within diverse cultural settings. To ensure that teachers' voices were well-represented at the Study Conference, we invited several practitioners actively involved in collaborative geometry projects to participate in a plenary panel that focused on French perspectives of teaching geometry.

We would like to thank the Local Organizing Committee members and their support staff and students of the l'IREM de Reims at the Université de Reims Champagne-Ardenne for all their work in hosting a successful ICMI Study 26 Conference. Preparation of the ICMI Study 26 volume, begun during the Study Conference, will continue in four virtual collaborative groups, during 2024-25. We aim for the Study Volume to be published in 2026 by Springer Nature in the New ICMI Studies Series.

An ICMI Study requires sustained commitment on three major projects (the Discussion Document, Study Conference and Study Volume) throughout its focused program of work. Along with the unwavering support and guidance we have received from Frederick Leung and Jean-Luc Dorier, we would like to acknowledge the help we have received from Merrilyn Goos (Vice President of ICMI), and Lena Koch (IMU Secretariat) at critical times during the development of the work program.

Angel Gutiérrez and Thomas Lowrie

26th ICMI Study co-chairs
D. Multidisciplinary perspectives.

These topics and related sub-topics provided the basis for inviting papers. Each submission was reviewed by at least two IPC members, and authors of accepted papers were invited to participate in the Study Conference. Accepted papers were then revised (when necessary) by the authors, before being published in these electronic conference proceedings. The conference proceedings include 60 papers from around the world. The countries represented in the Proceedings include: Argentina, Australia, Brazil, Canada, Chile, China, Colombia, Costa Rica, Croatia, France, Germany, Greece, Hong Kong, India, Iran, Israel, Italy, Japan, Malawi, Mexico, New Zealand, Portugal, Serbia, South Africa, Spain, Sweden, Switzerland, United Kingdom, and the United States.

In addition, we invited two esteemed scholars to present plenary lectures and another five to participate in a plenary panel on geometry practices within diverse cultural settings. To ensure that teachers' voices were well-represented at the Study Conference, we invited several practitioners actively involved in collaborative geometry projects to participate in a plenary panel that focused on French perspectives of teaching geometry.

We would like to thank the Local Organizing Committee members and their support staff and students of the l'IREM de Reims at the Université de Reims Champagne-Ardenne for all their work in hosting a successful ICMI Study 26 Conference. Preparation of the ICMI Study 26 volume, begun during the Study Conference, will continue in four virtual collaborative groups, during 2024-25. We aim for the Study Volume to be published in 2026 by Springer Nature in the New ICMI Studies Series.

An ICMI Study requires sustained commitment on three major projects (the Discussion Document, Study Conference and Study Volume) throughout its focused program of work. Along with the unwavering support and guidance we have received from Frederick Leung and Jean-Luc Dorier, we would like to acknowledge the help we have received from Merrilyn Goos (Vice President of ICMI), and Lena Koch (IMU Secretariat) at critical times during the development of the work program.

Angel Gutiérrez and Thomas Lowrie

26th ICMI Study co-chairs
PLENARY LECTURES
PLENARY LECTURES
RESEARCH ON TEACHING KNOWLEDGE IN GEOMETRY: THE CASE
OF PROOF IN THE UNITED STATES

Patricio Herbst

University of Michigan

High school geometry courses, at least in the United States, have historically been places where students could encounter important elements of mathematical practice–proofs in particular. But a quick look at this practice also attests to a cleavage between expert mathematical practice and school mathematical practice. For instructional improvement to bridge that cleavage (improving the curriculum or improving teachers), our understanding of existing instructional practice needs to be informed by more than a quick look. How can we describe the teaching knowledge implicit in the instructional practices of proof in geometry and how can this knowledge assist in the design and study of improvements in instruction?

In this lecture, I will describe a program of descriptive research that seeks to understand what we call the practical rationality of mathematics teaching and how this rationality can make room for instructional resources and practices that seek to improve instruction. The program of research on the practical rationality of mathematics teaching has had research on the teaching of mathematical proof in geometry as one of its core cases for empirical investigation. Hence, this presentation goes back and forth between relatively general theoretical ideas and specific research and development projects involving geometric proof.

I argue for a conceptualization of teaching knowledge that includes both explicit and tacit knowledge, knowledge that can be located in individuals as well as knowledge that is held collectively in the practice of teaching. I illustrate the first type of knowledge with examples of mathematical knowledge for teaching proof in geometry that include knowledge needed to manage instructional situations in geometry and knowledge needed to enact mathematical tasks of teaching. I illustrate the second type of knowledge with recognition of the norms of the situation of doing proofs in geometry. I will describe, in particular, how we measure teachers’ recognition of situational norms in geometry and how this recognition relates to measures of explicit mathematical knowledge for teaching geometry. The main results of this line of research are theoretical and methodological: We have been able to demonstrate that the practice of teaching geometry affords particular categories of perception that play a role in the teaching of proof in geometry and that teachers’ reactions to scenarios can be used to measure the extent to which teachers recognize those categories. Our work has also contributed to instructional improvement by uncovering possibilities for developing instructional resources and practices that support an enhanced role for proof in the study of geometry.
RESEARCH ON TEACHING KNOWLEDGE IN GEOMETRY: THE CASE
OF PROOF IN THE UNITED STATES

Patricio Herbst

University of Michigan

High school geometry courses, at least in the United States, have historically been places where students could encounter important elements of mathematical practice–proofs in particular. But a quick look at this practice also attests to a cleavage between expert mathematical practice and school mathematical practice. For instructional improvement to bridge that cleavage (improving the curriculum or improving teachers), our understanding of existing instructional practice needs to be informed by more than a quick look. How can we describe the teaching knowledge implicit in the instructional practices of proof in geometry and how can this knowledge assist in the design and study of improvements in instruction?

In this lecture, I will describe a program of descriptive research that seeks to understand what we call the practical rationality of mathematics teaching and how this rationality can make room for instructional resources and practices that seek to improve instruction. The program of research on the practical rationality of mathematics teaching has had research on the teaching of mathematical proof in geometry as one of its core cases for empirical investigation. Hence, this presentation goes back and forth between relatively general theoretical ideas and specific research and development projects involving geometric proof.

I argue for a conceptualization of teaching knowledge that includes both explicit and tacit knowledge, knowledge that can be located in individuals as well as knowledge that is held collectively in the practice of teaching. I illustrate the first type of knowledge with examples of mathematical knowledge for teaching proof in geometry that include knowledge needed to manage instructional situations in geometry and knowledge needed to enact mathematical tasks of teaching. I illustrate the second type of knowledge with recognition of the norms of the situation of doing proofs in geometry. I will describe, in particular, how we measure teachers’ recognition of situational norms in geometry and how this recognition relates to measures of explicit mathematical knowledge for teaching geometry. The main results of this line of research are theoretical and methodological: We have been able to demonstrate that the practice of teaching geometry affords particular categories of perception that play a role in the teaching of proof in geometry and that teachers’ reactions to scenarios can be used to measure the extent to which teachers recognize those categories. Our work has also contributed to instructional improvement by uncovering possibilities for developing instructional resources and practices that support an enhanced role for proof in the study of geometry.
TEACHING AND LEARNING GEOMETRY IN EARLY GRADES WITH
TECHNOLOGY

Nathalie Sinclair

Simon Fraser University, Canada

This plenary talk will be divided into three parts. In the first, I will provide an overview of the main findings of my research on the teaching and learning of geometry using of dynamic geometry software (specifically, The Geometer's Sketchpad) at the primary school level. This will include studies focussing on a number of different curricular concepts such as symmetry, angles, triangles and area involving students from 6 to 12 years old. I will discuss the specific affordances of Sketchpad that were relevant to these findings, as well as task design principles employed, and the specific theoretical constructs that were used to generate insights into the teaching and learning of geometry. In the second part, I will discuss this research in light of more contemporary perspectives. Indeed, since this research was carried out, for the most part, a decade ago, I am now in a position to reflect on how it could be conjugated with some new directions in the research literature, including recent work on spatial reasoning and the duo of artefact approach. Finally, in the third part of the talk, I will discuss more recent exploratory work involving new digital technologies (including 3d pens and the multitouch application The Griddler), and less Euclidean-based concepts (including 3d geometry, modelling and perspective geometry). I will use this work to propose some productive directions for future research in the teaching and learning of geometry using technology at the primary school level.
I will link my proposals to some recent theorising about the role of geometry in contemporary mathematics, about alternative curriculum approaches that challenge both Western and Piagetian progressions and about pedagogical approaches that centre making and acting.
TEACHING AND LEARNING GEOMETRY IN EARLY GRADES WITH
TECHNOLOGY

Nathalie Sinclair

Simon Fraser University, Canada

This plenary talk will be divided into three parts. In the first, I will provide an overview of the main findings of my research on the teaching and learning of geometry using of dynamic geometry software (specifically, The Geometer's Sketchpad) at the primary school level. This will include studies focussing on a number of different curricular concepts such as symmetry, angles, triangles and area involving students from 6 to 12 years old. I will discuss the specific affordances of Sketchpad that were relevant to these findings, as well as task design principles employed, and the specific theoretical constructs that were used to generate insights into the teaching and learning of geometry. In the second part, I will discuss this research in light of more contemporary perspectives. Indeed, since this research was carried out, for the most part, a decade ago, I am now in a position to reflect on how it could be conjugated with some new directions in the research literature, including recent work on spatial reasoning and the duo of artefact approach. Finally, in the third part of the talk, I will discuss more recent exploratory work involving new digital technologies (including 3d pens and the multitouch application The Griddler), and less Euclidean-based concepts (including 3d geometry, modelling and perspective geometry). I will use this work to propose some productive directions for future research in the teaching and learning of geometry using technology at the primary school level.
I will link my proposals to some recent theorising about the role of geometry in contemporary mathematics, about alternative curriculum approaches that challenge both Western and Piagetian progressions and about pedagogical approaches that centre making and acting.
PLENARY PANEL 1
PLENARY PANEL 1
# FRAMING GEOMETRIC REPRESENTATIONS AND PRACTICES IN CULTURALLY DIVERSE SETTINGS

Tom Lowrie

University of Canberra

*Our plenary panel will draw upon the experiences of mathematics educators from five countries, spanning four continents. The focus of the panel is to consider new directions for geometry education research framed with the political, cultural, and contextual dimensions of practice.*

## INTRODUCTION

One way to better understand the complexities of classroom practices in diverse contextual, cultural, and political contexts is to position mathematics education as a practice. Mathematics education practice differs across classrooms within a school but is likely more diverse across countries and continents. This plenary panel is initially framed within a theoretical stance that acknowledges that practice is informed by the social-political (relatings), cultural-discursive (sayings), and material-economic (doings) conditions of practice both from within the school community and across the broader professional context. These arrangements of practice (relatings, sayings and doings) are a mechanism for understanding how educational practices take place and are influenced by policies, the discourse of school communities, and the economic conditions of place and work (Practice Architectures, Kemmis, 2019; Kemmis & Grootenboer, 2008).

Relatings are shaped by the prefigured existing social orders and norms of schools and systems. It is acknowledged that the practice of education is conducted in and through relationships with others (teacher/student, teacher/teacher, teacher/principal, teacher/student's family). Sayings are shaped by culturally-discursive conditions that emerge from the 'living out' of traditions and theory. In a digital age, access to various forms of communication influence the specialized language of the practice of 'education' including how policies, rules, and curricula are reported. Doings are shaped by the economies that enable and constrain education, the materials available, and the settings where the practice of education takes place. In terms of doings, much of the work that takes place is influenced by the traditions of the settings and context.

## Framing geometry education in diverse settings

It could be argued that the school mathematics curriculum should reflect the political, cultural, and contextual contexts of the students for which it is planned. Political influences include the way power is achieved or used in a society. Increasingly, political influences involve the way politicians and the media provide personal views on how curricula should be shaped. Cultural practices include the shared values, attitudes, beliefs, and arrangements of a community. These practices can influence the language used in instruction or the degree to which informal and formal geometric knowledge is valued and represented.
# FRAMING GEOMETRIC REPRESENTATIONS AND PRACTICES IN CULTURALLY DIVERSE SETTINGS

Tom Lowrie

University of Canberra

*Our plenary panel will draw upon the experiences of mathematics educators from five countries, spanning four continents. The focus of the panel is to consider new directions for geometry education research framed with the political, cultural, and contextual dimensions of practice.*

## INTRODUCTION

One way to better understand the complexities of classroom practices in diverse contextual, cultural, and political contexts is to position mathematics education as a practice. Mathematics education practice differs across classrooms within a school but is likely more diverse across countries and continents. This plenary panel is initially framed within a theoretical stance that acknowledges that practice is informed by the social-political (relatings), cultural-discursive (sayings), and material-economic (doings) conditions of practice both from within the school community and across the broader professional context. These arrangements of practice (relatings, sayings and doings) are a mechanism for understanding how educational practices take place and are influenced by policies, the discourse of school communities, and the economic conditions of place and work (Practice Architectures, Kemmis, 2019; Kemmis & Grootenboer, 2008).

Relatings are shaped by the prefigured existing social orders and norms of schools and systems. It is acknowledged that the practice of education is conducted in and through relationships with others (teacher/student, teacher/teacher, teacher/principal, teacher/student's family). Sayings are shaped by culturally-discursive conditions that emerge from the 'living out' of traditions and theory. In a digital age, access to various forms of communication influence the specialized language of the practice of 'education' including how policies, rules, and curricula are reported. Doings are shaped by the economies that enable and constrain education, the materials available, and the settings where the practice of education takes place. In terms of doings, much of the work that takes place is influenced by the traditions of the settings and context.

## Framing geometry education in diverse settings

It could be argued that the school mathematics curriculum should reflect the political, cultural, and contextual contexts of the students for which it is planned. Political influences include the way power is achieved or used in a society. Increasingly, political influences involve the way politicians and the media provide personal views on how curricula should be shaped. Cultural practices include the shared values, attitudes, beliefs, and arrangements of a community. These practices can influence the language used in instruction or the degree to which informal and formal geometric knowledge is valued and represented.
Geometric representations and sense-making exist in all cultural groups and their meaning may differ from one culture to another.

## Making sense of practice(s) across settings

Cultural artifacts are objects created by members of distinct cultural groups that convey cultural meanings and information about their creators and users. Geometric ideas, procedures, techniques, and (indeed) practices are related to the development of cultural artifacts that are socio-culturally situated as well as distributed among these members from generation to generation. This approach also includes embodied cognition in which cognitive activities use symbols as external resources that assist these members to develop their mental representations and manipulation of objects.

## THEORETICAL AND METHODOLICAL CONSIDERATIONS

A Practice Architecture Framework provides a theoretical lens for understanding small-scale classroom and community initiatives as well as large-scale national agendas and curriculum reform (Lowrie, 2014). Practice architectures (Schatzki, 2002) are based on the premise that many practices are designed, especially the meta-practices associated with educational policy making and administration, curriculum development, and teaching and learning. Consequently, the relative fixed nature of facilities, equipment, resources, policy mandates, and curricula associated with specific subject areas all play a part in how practices are constructed, enabled, developed, and repurposed (Lowrie, 2014). To this point, practices are influenced by the context, culture, and infrastructure of the architecture that has been designed (see Figure 1). Consequently, practices are situated and shaped by the circumstances and conditions of the physical location in which they occur, and of course the time, and space of the circumstances.

<figure><img src="image_6.png" /><figcaption>Figure 1: Geometry performance by spatial level and strategy dominance</figcaption></figure>

This panel provides an opportunity to explore the diverse circumstances and conditions of spaces in classrooms and systems in Australasia, Africa, Asia, and South America.
Geometric representations and sense-making exist in all cultural groups and their meaning may differ from one culture to another.

## Making sense of practice(s) across settings

Cultural artifacts are objects created by members of distinct cultural groups that convey cultural meanings and information about their creators and users. Geometric ideas, procedures, techniques, and (indeed) practices are related to the development of cultural artifacts that are socio-culturally situated as well as distributed among these members from generation to generation. This approach also includes embodied cognition in which cognitive activities use symbols as external resources that assist these members to develop their mental representations and manipulation of objects.

## THEORETICAL AND METHODOLICAL CONSIDERATIONS

A Practice Architecture Framework provides a theoretical lens for understanding small-scale classroom and community initiatives as well as large-scale national agendas and curriculum reform (Lowrie, 2014). Practice architectures (Schatzki, 2002) are based on the premise that many practices are designed, especially the meta-practices associated with educational policy making and administration, curriculum development, and teaching and learning. Consequently, the relative fixed nature of facilities, equipment, resources, policy mandates, and curricula associated with specific subject areas all play a part in how practices are constructed, enabled, developed, and repurposed (Lowrie, 2014). To this point, practices are influenced by the context, culture, and infrastructure of the architecture that has been designed (see Figure 1). Consequently, practices are situated and shaped by the circumstances and conditions of the physical location in which they occur, and of course the time, and space of the circumstances.

<figure><img src="image_5.png" /><figcaption>Figure 1: Geometry performance by spatial level and strategy dominance</figcaption></figure>

This panel provides an opportunity to explore the diverse circumstances and conditions of spaces in classrooms and systems in Australasia, Africa, Asia, and South America.
An ethnomathematics methodology (Rosa & Orey, 2016) will be used to examine and make sense of the mathematical and geometrical knowledge (ideas, processes, and practices) that originate from diverse cultural contexts through history. The ethnomathematics methodology is multi-dimensional and provides a broad scope of affordances, namely: cognitive; conceptual; educational; epistemological; historical; and political. According to Rosa and Orey (2016) these dimensions are interrelated and aim to analyze sociocultural roots of mathematical knowledge. Moreover, the theoretical underpinnings of practice architectures, and the methodological tools established within ethnomathematics, align well and provide synergies for a rich panel discussion. Ethnomathematics is also a program that studies geometric concepts and theories developed locally as well as it is related to dynamic pedagogical actions that respond to the environmental, social, cultural, political, and economic needs of the students, which enables them to develop their imagination and creativity.

## Questions to be consider by the panelists

* How can socio-political and cultural contexts of mathematics curricula help to recognize and respect the history, tradition, and mathematical thinking developed by members of distinct cultural groups through the development of geometric thinking?

* What is the significance of the influence on the curriculum of different political and cultural contexts for researchers, policy makers, and teachers?

* What cultural traits contribute to the construction of local geometric knowledge?

* What role does ethnomathematics play in the understanding of diverse indigenous ways of doing geometry?

* What are the sociocultural influences of the use of this knowledge in distinct cultural groups?

## DISCUSSION

The forthcoming papers describe the complex interactions between practice arrangements within distinct contexts in Africa (Malawi), Asia (Iran), and South America (Argentina and Brazil). The cultural traditions, political and social constructs, and the economic arrangements of these countries differ dramatically, even with continents. Nevertheless, some common tensions endure across these cases, particularly in relation to tensions that exist between formal (represented) school geometry and meaningful geometric practices that play out in lived contexts. Geometry education has a sustained and important history in these contexts, however, culturally-meaningful practices are not necessarily valued.

It may be the case that the global (and dominant) understanding of geometry education is too narrow, and consequently, it undoes the relationship between knowing and practicing. The following four cases propose that geometry practices should value cultural and contextual traditions in order to flourish and sustain communities.

## References

Kemmis, S. (2019). *A practice sensibility: An Invitation to the Theory of Practice Architectures*. Singapore: Springer Nature.

Kemmis, S., and P. Grootenboer. (2008). Situating Praxis in Practice: Practice Architectures and the Cultural, Social and Material Conditions for Practice. In S. Kemmis and T. J. Smith (Eds.) *Enabling Praxis: Challenges for Education*. Rotterdam: Sense Publishers.
An ethnomathematics methodology (Rosa & Orey, 2016) will be used to examine and make sense of the mathematical and geometrical knowledge (ideas, processes, and practices) that originate from diverse cultural contexts through history. The ethnomathematics methodology is multi-dimensional and provides a broad scope of affordances, namely: cognitive; conceptual; educational; epistemological; historical; and political. According to Rosa and Orey (2016) these dimensions are interrelated and aim to analyze sociocultural roots of mathematical knowledge. Moreover, the theoretical underpinnings of practice architectures, and the methodological tools established within ethnomathematics, align well and provide synergies for a rich panel discussion. Ethnomathematics is also a program that studies geometric concepts and theories developed locally as well as it is related to dynamic pedagogical actions that respond to the environmental, social, cultural, political, and economic needs of the students, which enables them to develop their imagination and creativity.

### Questions to be consider by the panelists

* How can socio-political and cultural contexts of mathematics curricula help to recognize and respect the history, tradition, and mathematical thinking developed by members of distinct cultural groups through the development of geometric thinking?

* What is the significance of the influence on the curriculum of different political and cultural contexts for researchers, policy makers, and teachers?

* What cultural traits contribute to the construction of local geometric knowledge?

* What role does ethnomathematics play in the understanding of diverse indigenous ways of doing geometry?

* What are the sociocultural influences of the use of this knowledge in distinct cultural groups?

## DISCUSSION

The forthcoming papers describe the complex interactions between practice arrangements within distinct contexts in Africa (Malawi), Asia (Iran), and South America (Argentina and Brazil). The cultural traditions, political and social constructs, and the economic arrangements of these countries differ dramatically, even with continents. Nevertheless, some common tensions endure across these cases, particularly in relation to tensions that exist between formal (represented) school geometry and meaningful geometric practices that play out in lived contexts. Geometry education has a sustained and important history in these contexts, however, culturally-meaningful practices are not necessarily valued.

It may be the case that the global (and dominant) understanding of geometry education is too narrow, and consequently, it undoes the relationship between knowing and practicing. The following four cases propose that geometry practices should value cultural and contextual traditions in order to flourish and sustain communities.

## References

Kemmis, S. (2019). *A practice sensibility: An Invitation to the Theory of Practice Architectures*. Singapore: Springer Nature.

Kemmis, S., and P. Grootenboer. (2008). Situating Praxis in Practice: Practice Architectures and the Cultural, Social and Material Conditions for Practice. In S. Kemmis and T. J. Smith (Eds.) *Enabling Praxis: Challenges for Education*. Rotterdam: Sense Publishers.
Lowrie, T. (2014). An educational practices framework: The potential for empowerment of the teaching profession. *Journal of Education for Teaching, 40*(1), 34-46.

Orey, D. C., & Rosa, M. (2021). Ethnomodelling as a glocalization process of mathematical practices through cultural dynamism. *The Mathematics Enthusiast, 18*(3), 438-468.

Rosa, M., Orey, D.C. (2016). State of the Art in Ethnomathematics. In: *Current and Future Perspectives of Ethnomathematics as a Program*. ICME-13 Topical Surveys. Springer, Cham. https://doi.org/10.1007/978-3-319-30120-4_3.

Schatzki, T. R. (2002). *The Site of the Social: A Philosophical Account of the Constitution of Social Life and Change*. University Park: University of Pennsylvania Press.
Lowrie, T. (2014). An educational practices framework: The potential for empowerment of the teaching profession. *Journal of Education for Teaching, 40*(1), 34-46.

Orey, D. C., & Rosa, M. (2021). Ethnomodelling as a glocalization process of mathematical practices through cultural dynamism. *The Mathematics Enthusiast, 18*(3), 438-468.

Rosa, M., Orey, D.C. (2016). State of the Art in Ethnomathematics. In: *Current and Future Perspectives of Ethnomathematics as a Program*. ICME-13 Topical Surveys. Springer, Cham. https://doi.org/10.1007/978-3-319-30120-4_3.

Schatzki, T. R. (2002). *The Site of the Social: A Philosophical Account of the Constitution of Social Life and Change*. University Park: University of Pennsylvania Press.
# CONTEXT AND LANGUAGE IN MALAWI CLASSROOMS

Lisnet Mwadzaangati

University of Malawi

## THE CONTEXT OF MATHEMATICS EDUCATION IN MALAWI

Malawi gained independence from a British colony to an authoritarian political regime in 1964 and later to a democratic regime in 1994. The difference on type and quality of education prior to and during the authoritarian political regime was very little as there were no spaces for negotiating education reform (Mwakapenda, 2002). From 1994, the major reforms in education began to take place, including the introduction of free primary education and a curriculum review by the Malawi government through the Ministry of Education (MoE). The introduction of free primary education led to a massive increase in enrolment at primary school, which resulted in an increase in secondary education to an extent that the Malawi government converted Malawi College of Distance Education (MCDE) centres into community day secondary schools (CDSSs) in 1998 (Malawi Government, 2008). The increase in access to primary and secondary education exerted enormous pressure on the education infrastructure, learning materials, and education human resources. In primary school, there was a problem of teaching resources, infrastructure, and lack of teachers because many experienced teachers were deployed to teach at the CDSSs. In the secondary schools, the challenge was a lack of teaching resources and lack of qualified teachers in quality and quantity. The challenge of quality rose because the teachers who were deployed into CDSSs were not trained thoroughly while the challenge of quantity rose because the government did not deploy enough teachers to the CDSSs.

Despite making strides in achieving education access through introduction of free primary education, education quality remains a challenge as the Malawi education system continues to grapple with the effects of introduction of free primary education as well as the rapid increase in population growth (MoE, 2023). The deteriorating of education quality perpetuates learners' poor performance at both national and international assessments in literacy and numeracy at both local and international primary education level assessments (Milner et al., 2011), which in return led to students' low performance in mathematics at secondary level national examinations (Ministry of Education, Science and Technology [MoEST], 2020).

Several education reforms have taken place in all subjects and at all education levels since 1994 with an aim of improving education quality and democratising education (Mwakapenda, 2002). In 2006, the Malawi government shifted from objective education model (OEM) to outcome-based education model (OBE) with underlying argument that OEM was teacher centred, hence teachers played a more active role in achieving learning objectives than the learners (MoE, 2006). The assumption was that OBE enhances education quality and students' performance by promoting students' active participation during instruction and sustaining independent learning after instruction (Chang & Salalahi, 2017). Although the curriculum review was accompanied by several other interventions to improve education quality such as training of more primary and secondary school teachers and building more classrooms, the teaching and learning of mathematics (especially geometry) remains a
# CONTEXT AND LANGUAGE IN MALAWI CLASSROOMS

Lisnet Mwadzaangati

University of Malawi

## THE CONTEXT OF MATHEMATICS EDUCATION IN MALAWI

Malawi gained independence from a British colony to an authoritarian political regime in 1964 and later to a democratic regime in 1994. The difference on type and quality of education prior to and during the authoritarian political regime was very little as there were no spaces for negotiating education reform (Mwakapenda, 2002). From 1994, the major reforms in education began to take place, including the introduction of free primary education and a curriculum review by the Malawi government through the Ministry of Education (MoE). The introduction of free primary education led to a massive increase in enrolment at primary school, which resulted in an increase in secondary education to an extent that the Malawi government converted Malawi College of Distance Education (MCDE) centres into community day secondary schools (CDSSs) in 1998 (Malawi Government, 2008). The increase in access to primary and secondary education exerted enormous pressure on the education infrastructure, learning materials, and education human resources. In primary school, there was a problem of teaching resources, infrastructure, and lack of teachers because many experienced teachers were deployed to teach at the CDSSs. In the secondary schools, the challenge was a lack of teaching resources and lack of qualified teachers in quality and quantity. The challenge of quality rose because the teachers who were deployed into CDSSs were not trained thoroughly while the challenge of quantity rose because the government did not deploy enough teachers to the CDSSs.

Despite making strides in achieving education access through introduction of free primary education, education quality remains a challenge as the Malawi education system continues to grapple with the effects of introduction of free primary education as well as the rapid increase in population growth (MoE, 2023). The deteriorating of education quality perpetuates learners' poor performance at both national and international assessments in literacy and numeracy at both local and international primary education level assessments (Milner et al., 2011), which in return led to students' low performance in mathematics at secondary level national examinations (Ministry of Education, Science and Technology [MoEST], 2020).

Several education reforms have taken place in all subjects and at all education levels since 1994 with an aim of improving education quality and democratising education (Mwakapenda, 2002). In 2006, the Malawi government shifted from objective education model (OEM) to outcome-based education model (OBE) with underlying argument that OEM was teacher centred, hence teachers played a more active role in achieving learning objectives than the learners (MoE, 2006). The assumption was that OBE enhances education quality and students' performance by promoting students' active participation during instruction and sustaining independent learning after instruction (Chang & Salalahi, 2017). Although the curriculum review was accompanied by several other interventions to improve education quality such as training of more primary and secondary school teachers and building more classrooms, the teaching and learning of mathematics (especially geometry) remains a
challenge (MoEST, 2020). The main contributing factor being the lack of qualified teachers. The Malawi education statistics reports that only 10 percent of the secondary school teachers are qualified to teach mathematics (MoE, 2023). This implies persistence of two worrisome challenges on development of mathematics education in Malawi; (1) access to secondary mathematics education is limited in Malawi, and (2) low quality of mathematics education in Malawi, hence limiting students. This is in contradiction with Malawi's aspirations that the quality of education has to be at par with international standards in an increasingly knowledge based world (MoE, 2023). This means that mathematics education will continue to be a challenge if there are no concerted and consolidated efforts amongst the education stakeholders to address the teacher knowledge issues.

To achieve learner centred education and education quality in Malawi, teachers continue to be urged to move from traditional teaching methods, which are characterized as ‘chalk and talk’, to more inquiry-based learning characterized as ‘student-centred’ (MoEST, 2020). However, due to the challenges of teacher knowledge, large class sizes, and very few resources for schools, most classrooms in Malawi continue to be dominated by teacher talk, hence learners find the learning of mathematics hard (Phiri, 2018). Both Malawian primary and secondary school class sizes can average 100 learners especially in CDSSs. Nevertheless, the teachers do try to use different student centred strategies including discussions in small groups where students learn from and support each other using carefully designed hands-on teaching activities. This is evidenced by Malemya (2018) who reports that Malawian mathematics classroom instruction includes the following characteristics:

- combines both learner-centred and teacher centred approaches where not much is done to help learners deeply understand the concepts to be able to independently develop thinking capabilities, relies on a textbook, focuses on developing a mathematical skill, devotes most available time to practising routine procedures, features isolated tasks, there is more listening from a teacher talking, relatively large classes. (p. 171)

These characteristics point to the challenge of knowledge of mathematical language use in Malawian classrooms, also called language responsive teaching as evidenced by some researchers in secondary geometry teaching in Malawi (Adler, 2022; Planas et al., 2022).

## LANGUAGE AND THE TEACHING OF GEOMETRY IN MALAWI

Some of the reforms that have been made in the education sector since Malawi got her independence in 1964 have been in the area of language education. The education system in Malawi comprises 8 years of primary education, (Standard 1 to Standard 8), 4 years of secondary education (Form 1 to 4) and 4 years of university education year 1 to 4). There are over 15 local languages in Malawi, but Chichewa remains the Malawi's national language, which is used as a mandatory medium of instruction from standards (grade) 1 to 4 in all public primary schools. Since 1989, all pupils' books (except those for English studies) for standards 1 to 4 are written in Chichewa (Kunje, 2009; Malemya, 2018). Although the language policy changed in 1996, instructing that the language of instruction from standards 1 to 4 be the pupils' mother tongue, classroom evidence reveal that Chichewa remains the language that children learn mathematics even in areas where it is not dominantly spoken, probably lack of support to teachers in the form of mother tongue training and development of mother tongue instructional material (Kunje et al., 2009). This is of great worry because learning in unfamiliar language especially during early years implies learning the language itself and the content of the subject being studied at the same time.
challenge (MoEST, 2020). The main contributing factor being the lack of qualified teachers. The Malawi education statistics reports that only 10 percent of the secondary school teachers are qualified to teach mathematics (MoE, 2023). This implies persistence of two worrisome challenges on development of mathematics education in Malawi; (1) access to secondary mathematics education is limited in Malawi, and (2) low quality of mathematics education in Malawi, hence limiting students. This is in contradiction with Malawi's aspirations that the quality of education has to be at par with international standards in an increasingly knowledge based world (MoE, 2023). This means that mathematics education will continue to be a challenge if there are no concerted and consolidated efforts amongst the education stakeholders to address the teacher knowledge issues.

To achieve learner centred education and education quality in Malawi, teachers continue to be urged to move from traditional teaching methods, which are characterized as ‘chalk and talk’, to more inquiry-based learning characterized as ‘student-centred’ (MoEST, 2020). However, due to the challenges of teacher knowledge, large class sizes, and very few resources for schools, most classrooms in Malawi continue to be dominated by teacher talk, hence learners find the learning of mathematics hard (Phiri, 2018). Both Malawian primary and secondary school class sizes can average 100 learners especially in CDSSs. Nevertheless, the teachers do try to use different student centred strategies including discussions in small groups where students learn from and support each other using carefully designed hands-on teaching activities. This is evidenced by Malemya (2018) who reports that Malawian mathematics classroom instruction includes the following characteristics:

- combines both learner-centred and teacher centred approaches where not much is done to help learners deeply understand the concepts to be able to independently develop thinking capabilities, relies on a textbook, focuses on developing a mathematical skill, devotes most available time to practising routine procedures, features isolated tasks, there is more listening from a teacher talking, relatively large classes. (p. 171)

These characteristics point to the challenge of knowledge of mathematical language use in Malawian classrooms, also called language responsive teaching as evidenced by some researchers in secondary geometry teaching in Malawi (Adler, 2022; Planas et al., 2022).

## LANGUAGE AND THE TEACHING OF GEOMETRY IN MALAWI

Some of the reforms that have been made in the education sector since Malawi got her independence in 1964 have been in the area of language education. The education system in Malawi comprises 8 years of primary education, (Standard 1 to Standard 8), 4 years of secondary education (Form 1 to 4) and 4 years of university education year 1 to 4). There are over 15 local languages in Malawi, but Chichewa remains the Malawi's national language, which is used as a mandatory medium of instruction from standards (grade) 1 to 4 in all public primary schools. Since 1989, all pupils' books (except those for English studies) for standards 1 to 4 are written in Chichewa (Kunje, 2009; Malemya, 2018). Although the language policy changed in 1996, instructing that the language of instruction from standards 1 to 4 be the pupils' mother tongue, classroom evidence reveal that Chichewa remains the language that children learn mathematics even in areas where it is not dominantly spoken, probably lack of support to teachers in the form of mother tongue training and development of mother tongue instructional material (Kunje et al., 2009). This is of great worry because learning in unfamiliar language especially during early years implies learning the language itself and the content of the subject being studied at the same time.
From standard 5, the language policy dictates that instruction should be in English, which remains
Malawi's official language even in the postcolonial period. As such, all curriculum materials from
standard 5 up to secondary level mathematics are in English, including the syllabus and all textbooks.
Unlike in other developed countries where the language of instruction is the language of the majority,
in Malawi, the majority of learners speak a different language than the language of instruction. There
is rich literature that argues for the benefit of engaging learners in rich mathematical discourse
practices especially for learners in multilingual classrooms to enable them to access the mathematics
register, ways of speaking, reading and writing mathematics, which emphasizes mathematical
reasoning and explanations (e.g., Prediger, 2019).

Both the Malawian primary and secondary geometry entails a relatively formal, static, and deductive approaches. The routine approach to secondary geometry starts from definitions to illustrations of geometric concepts, and then examples for practicing and applying the geometric concepts in usually what is called dialogic instruction. For some teachers who embrace learner-centred education related values, and are concerned with learner involvement, there is use of everyday examples, concrete objects and diagrams for empirical measurement tasks, which are suggested in curriculum materials such as the syllabus and textbooks (Mwadzaangati et al., 2022). However, there remains a challenge of engaging learners in discussing and exploring of meanings even in classrooms where these learner-centred approaches are practiced (Adler et al., 2022). Consequently, the deductive approach prolongs the well-known student difficulties in geometry and obscures them from working simultaneously and in connected ways with the visual and verbal representations of geometric objects (See Bansilal & Ubah, 2019).

Seago et al. (2014) has argued that although presenting of both static approach and the dynamic approaches (in their case the transformations-based approach to the teaching of similarity) to the teaching of geometry are mathematically precise, the dynamic approach provides learners with robust learning opportunities because it offers more clarity to geometric concepts than the static approach. In Mwadzaangati (2019), I have argued that while dynamic approaches to geometry are considered to be effective in enhancing learners understanding of geometric concepts, its conceptualization is not possible in developing countries like Malawi, which are characterized by dire lack of teaching and learning resources. As such the Malawian curriculum recourses such as textbooks are designed in a way that can enable teachers and learners to use readily accessible resources to them such as pencil and paper. Furthermore, there are attempts in some of the textbooks to include empirical exploration tasks that can serve as attempts to address limitations of pencil and paper tasks, hence the instructional approach used in the textbook, which present some proof problems as experimental problems and then as a formal proof, might help the learners to both discuss and explore inductive and deductive reasoning geometric process (Mwadzaangati, 2019). This means that the challenges for mathematics researchers, educators, and even policy makers in addressing the teachers' challenges in geometry teaching in resource constrained contexts like Malawi require strategic actions of dealing with how to teach geometry effectively using static approaches. Thus, how can we help the teachers to promote learners discussions and explorations using the readily available recourses (e.g., paper and pencil)?
From standard 5, the language policy dictates that instruction should be in English, which remains
Malawi's official language even in the postcolonial period. As such, all curriculum materials from
standard 5 up to secondary level mathematics are in English, including the syllabus and all textbooks.
Unlike in other developed countries where the language of instruction is the language of the majority,
in Malawi, the majority of learners speak a different language than the language of instruction. There
is rich literature that argues for the benefit of engaging learners in rich mathematical discourse
practices especially for learners in multilingual classrooms to enable them to access the mathematics
register, ways of speaking, reading and writing mathematics, which emphasizes mathematical
reasoning and explanations (e.g., Prediger, 2019).

Both the Malawian primary and secondary geometry entails a relatively formal, static, and deductive approaches. The routine approach to secondary geometry starts from definitions to illustrations of geometric concepts, and then examples for practicing and applying the geometric concepts in usually what is called dialogic instruction. For some teachers who embrace learner-centred education related values, and are concerned with learner involvement, there is use of everyday examples, concrete objects and diagrams for empirical measurement tasks, which are suggested in curriculum materials such as the syllabus and textbooks (Mwadzaangati et al., 2022). However, there remains a challenge of engaging learners in discussing and exploring of meanings even in classrooms where these learner-centred approaches are practiced (Adler et al., 2022). Consequently, the deductive approach prolongs the well-known student difficulties in geometry and obscures them from working simultaneously and in connected ways with the visual and verbal representations of geometric objects (See Bansilal & Ubah, 2019).

Seago et al. (2014) has argued that although presenting of both static approach and the dynamic approaches (in their case the transformations-based approach to the teaching of similarity) to the teaching of geometry are mathematically precise, the dynamic approach provides learners with robust learning opportunities because it offers more clarity to geometric concepts than the static approach. In Mwadzaangati (2019), I have argued that while dynamic approaches to geometry are considered to be effective in enhancing learners understanding of geometric concepts, its conceptualization is not possible in developing countries like Malawi, which are characterized by dire lack of teaching and learning resources. As such the Malawian curriculum recourses such as textbooks are designed in a way that can enable teachers and learners to use readily accessible resources to them such as pencil and paper. Furthermore, there are attempts in some of the textbooks to include empirical exploration tasks that can serve as attempts to address limitations of pencil and paper tasks, hence the instructional approach used in the textbook, which present some proof problems as experimental problems and then as a formal proof, might help the learners to both discuss and explore inductive and deductive reasoning geometric process (Mwadzaangati, 2019). This means that the challenges for mathematics researchers, educators, and even policy makers in addressing the teachers' challenges in geometry teaching in resource constrained contexts like Malawi require strategic actions of dealing with how to teach geometry effectively using static approaches. Thus, how can we help the teachers to promote learners discussions and explorations using the readily available recourses (e.g., paper and pencil)?
References

Adler, J., Mwadzaangati, L., & Takker, S. (2023). From defining as assertion to defining as explaining meaning: teachers' learning through theory-informed lesson study. *International Journal for Lesson & Learning Studies, 12*(1), 38–51.

Bansilal, S., & Ubah, I. (2019). The use of semiotic representations in reasoning about similar triangles in Euclidean geometry. *Pythagoras, 40*(1), 1–10.

Chang, C. C., & Silalahi, S. M. (2017). A review and content analysis of mathematics textbooks in educational research. *Problems of Education in the 21st Century, 75*(3), 235–251.

Malawi Government, (2008). *Malawi Poverty Reduction Strategy Paper [MPRSP]*: Ministry of Finance: Lilongwe.

Kunje, D., Selemani-Meke, E., & Ogawa, K. (2009). An investigation of the relationship between school and pupil characteristics and achievement at the basic education level in Malawi. *Journal of International Cooperation in Education, 12*, 33–49.

Milner, G.D., Mulera, D. & Chimuzu, T. 2011. *The SACMEQ III project in Malawi: A study of the conditions of schooling and the quality of education*. Available at [http://www.sacmeq.org/?q=sacmeq-projects/sacmeq-iii/reports](http://www.sacmeq.org/?q=sacmeq-projects/sacmeq-iii/reports). Accessed 21 November 2019.

MoE, (2006). *Malawi, primary education curriculum reform and assessment framework*. Ministry of education: Lilongwe.

MoE. (2023). *Malawi education statistics report: Education management information system report*. Ministry of Education: Lilongwe.

MoEST, (2020). *National Education Sector Investment Plan 2020–2030*. Ministry of Education, Science and Technology.

Mwadzaangati, L. (2019). Comparison of geometric proof development tasks as set up in the textbook and as implemented by teachers in the classroom. *Pythagoras, 40*(1), 1–14.
[https://doi.org/%2010.4102/pythagoras.v40i1.458](https://doi.org/%2010.4102/pythagoras.v40i1.458)

Mwadzaangati, L., Adler, J., & Kazima, M. (2022). Mathematics mediational means and learner centredness: Insights from ‘traditional’ Malawian Secondary School Geometry Lessons. *African Journal of Research in Mathematics, Science and Technology Education, 1*–14. https://doi.org/10.1080/18117295.2022.2055910

Mwakapenda W. (2002). The Status and Context of Change in Mathematics Education in Malawi: *Educational Studies in Mathematics Journal* 49, 251–281.

Planas, N., Adler, J., & Mwadzaangati, L. (2022). What is mathematics teaching talk for? A response based on three sites of practice in mathematics education. *ZDM–Mathematics Education, 1*–14.
[https://doi.org/10.1007/s11858-022-01452-5](https://doi.org/10.1007/s11858-022-01452-5)

Prediger, S. (2019). Investigating and promoting teachers’ expertise for language-responsive mathematics teaching. *Mathematics Education Research Journal, 31*(4), 367–392. https://doi.org/10.1007/s13394-019-00258-1

Seago, N. M., Jacobs, J. K., Heck, D. J., Nelson, C. L., & Malzahn, K. A. (2014). Impacting teachers’ understanding of geometric similarity: Results from field testing of the Learning and Teaching Geometry professional development materials. *Professional Development in Education, 40*(4), 627–653.
https://doi.org/10.1080/19415257.2013.830144
## References

Adler, J., Mwadzaangati, L., & Takker, S. (2023). From defining as assertion to defining as explaining meaning: teachers' learning through theory-informed lesson study. *International Journal for Lesson & Learning Studies, 12*(1), 38–51.

Bansilal, S., & Ubah, I. (2019). The use of semiotic representations in reasoning about similar triangles in Euclidean geometry. *Pythagoras, 40*(1), 1–10.

Chang, C. C., & Silalahi, S. M. (2017). A review and content analysis of mathematics textbooks in educational research. *Problems of Education in the 21st Century, 75*(3), 235–251.

Malawi Government, (2008). *Malawi Poverty Reduction Strategy Paper [MPRSP]*: Ministry of Finance: Lilongwe.

Kunje, D., Selemani-Meke, E., & Ogawa, K. (2009). An investigation of the relationship between school and pupil characteristics and achievement at the basic education level in Malawi. *Journal of International Cooperation in Education, 12*, 33–49.

Milner, G.D., Mulera, D. & Chimuzu, T. 2011. *The SACMEQ III project in Malawi: A study of the conditions of schooling and the quality of education*. Available at [http://www.sacmeq.org/?q=sacmeq-projects/sacmeq-iii/reports](http://www.sacmeq.org/?q=sacmeq-projects/sacmeq-iii/reports). Accessed 21 November 2019.

MoE, (2006). *Malawi, primary education curriculum reform and assessment framework*. Ministry of education: Lilongwe.

MoE. (2023). *Malawi education statistics report: Education management information system report*. Ministry of Education: Lilongwe.

MoEST, (2020). *National Education Sector Investment Plan 2020–2030*. Ministry of Education, Science and Technology.

Mwadzaangati, L. (2019). Comparison of geometric proof development tasks as set up in the textbook and as implemented by teachers in the classroom. *Pythagoras, 40*(1), 1–14.
[https://doi.org/%2010.4102/pythagoras.v40i1.458](https://doi.org/%2010.4102/pythagoras.v40i1.458)

Mwadzaangati, L., Adler, J., & Kazima, M. (2022). Mathematics mediational means and learner centredness: Insights from ‘traditional’ Malawian Secondary School Geometry Lessons. *African Journal of Research in Mathematics, Science and Technology Education, 1*–14. [https://doi.org/10.1080/18117295.2022.2055910](https://doi.org/10.1080/18117295.2022.2055910)

Mwakapenda W. (2002). The Status and Context of Change in Mathematics Education in Malawi: *Educational Studies in Mathematics Journal* 49, 251–281.

Planas, N., Adler, J., & Mwadzaangati, L. (2022). What is mathematics teaching talk for? A response based on three sites of practice in mathematics education. *ZDM–Mathematics Education, 1*–14.
[https://doi.org/10.1007/s11858-022-01452-5](https://doi.org/10.1007/s11858-022-01452-5)

Prediger, S. (2019). Investigating and promoting teachers’ expertise for language-responsive mathematics teaching. *Mathematics Education Research Journal, 31*(4), 367–392. [https://doi.org/10.1007/s13394-019-00258-1](https://doi.org/10.1007/s13394-019-00258-1)

Seago, N. M., Jacobs, J. K., Heck, D. J., Nelson, C. L., & Malzahn, K. A. (2014). Impacting teachers’ understanding of geometric similarity: Results from field testing of the Learning and Teaching Geometry professional development materials. *Professional Development in Education, 40*(4), 627–653.
[https://doi.org/10.1080/19415257.2013.830144](https://doi.org/10.1080/19415257.2013.830144)
# FRAMING GEOMETRIC REPRESENTATIONS AND PRACTICES
## IN CULTURALLY-DIVERSE SETTINGS

Natalia Sgreccia

UNR-CONICET, Argentina

## INTRODUCTION

In Argentina, compulsory education covers children and young people from 4 to 18 years of age. In the province of Santa Fe, these 14 years are divided into 2 years of Initial Education, 7 years of Primary Education and 5 years of Secondary Education (6 years in technical-vocational schools). At the curricular level, the mathematics area is present in all these years (and even in numerous higher-level courses: university and tertiary). With respect to geometry education, compulsory education deals with geometric, linear, plane, and spatial forms, their characteristics and properties, providing a connection with aspects of daily life while training in observation, visualization, representation, estimation and deduction, among other mental operations. Measurements of different magnitudes are also developed, with their systems of units, conversions, and procedures.

The Ministry of Education of the Nation is responsible for guaranteeing comprehensive, permanent, and quality education for all the inhabitants of the country, setting the educational policy and controlling its compliance. The Ministry of Education of the Province of Santa Fe, in concert and concurrently with the first one, is responsible for planning, organization, supervision and financing, and guaranteeing access.

For the coordinated planning and academic organization of educational actions, and in harmony with the Priority Learning Nuclei at the national level, the provincial Ministry prepares Curriculum Designs in which it establishes general guidelines for teaching, as well as specifications of topics to be developed in each area along with methodological considerations. Focusing on Primary and Secondary Education, the topics are organized into four axes: Numbers and Operations; Geometry and Measurement; Statistics and Probability; Algebra and Functions.

## Geometry and Measurement in Argentina's Curricula

The Geometry component includes the study of geometric figures (non-empty sets whose elements are points) in different dimensions—including points, lines, segments, curves, planes, angles, polygons, circles, polyhedra, and round bodies. In addition, students study geometric places, conic sections, and vectors. Initial understandings involve synthetic approaches associated with the *geometry of shapes* in primary school and the basic cycle of secondary school, where some simple demonstrations can be carried out, but very formative in reasoning, to be integrated with the analytical approach *geometry of coordinates* in the last section of secondary school. Classifications of geometric figures are made, taking into account their constituent elements, shapes, and properties, and geometric constructions are carried out. For example, drawings of figures with ruler and compass, which consolidates the importance of geometric invariants and criteria to consider. Relationships between
# FRAMING GEOMETRIC REPRESENTATIONS AND PRACTICES
## IN CULTURALLY-DIVERSE SETTINGS

Natalia Sgreccia

UNR-CONICET, Argentina

## INTRODUCTION

In Argentina, compulsory education covers children and young people from 4 to 18 years of age. In the province of Santa Fe, these 14 years are divided into 2 years of Initial Education, 7 years of Primary Education and 5 years of Secondary Education (6 years in technical-vocational schools). At the curricular level, the mathematics area is present in all these years (and even in numerous higher-level courses: university and tertiary). With respect to geometry education, compulsory education deals with geometric, linear, plane, and spatial forms, their characteristics and properties, providing a connection with aspects of daily life while training in observation, visualization, representation, estimation and deduction, among other mental operations. Measurements of different magnitudes are also developed, with their systems of units, conversions, and procedures.

The Ministry of Education of the Nation is responsible for guaranteeing comprehensive, permanent, and quality education for all the inhabitants of the country, setting the educational policy and controlling its compliance. The Ministry of Education of the Province of Santa Fe, in concert and concurrently with the first one, is responsible for planning, organization, supervision and financing, and guaranteeing access.

For the coordinated planning and academic organization of educational actions, and in harmony with the Priority Learning Nuclei at the national level, the provincial Ministry prepares Curriculum Designs in which it establishes general guidelines for teaching, as well as specifications of topics to be developed in each area along with methodological considerations. Focusing on Primary and Secondary Education, the topics are organized into four axes: Numbers and Operations; Geometry and Measurement; Statistics and Probability; Algebra and Functions.

## Geometry and Measurement in Argentina's Curricula

The Geometry component includes the study of geometric figures (non-empty sets whose elements are points) in different dimensions—including points, lines, segments, curves, planes, angles, polygons, circles, polyhedra, and round bodies. In addition, students study geometric places, conic sections, and vectors. Initial understandings involve synthetic approaches associated with the *geometry of shapes* in primary school and the basic cycle of secondary school, where some simple demonstrations can be carried out, but very formative in reasoning, to be integrated with the analytical approach *geometry of coordinates* in the last section of secondary school. Classifications of geometric figures are made, taking into account their constituent elements, shapes, and properties, and geometric constructions are carried out. For example, drawings of figures with ruler and compass, which consolidates the importance of geometric invariants and criteria to consider. Relationships between
geometric “objects” are also analyzed—parallelism and perpendicularity of lines; tangency between curves; adjacency of angles; and congruence and similarity of figures.

The Measurement part could be a separate axis since it has its own identity and sufficient relevance due to its direct applicability in numerous situations of daily life. Measurement systems are studied and students learn to measure time, angles, length, area, volume, capacity, both directly using some unit of measurement to measure objects; and indirectly, through the use of formulas and arithmetic operations. It is therefore an interesting space for integrating topics. For example, geometry with the field of numbers (to calculate the volume of a cylinder, two geometric elements must be identified that define its shape and size—the radius of the base r and the height h—and its measurements for the calculation Volume=$π*r^2*h$). The notion of independence of the measurements of perimeter, area and volume is worked on, analyzing concrete and simple situations of non-linearity (if the area of one figure is equal to twice the area of another, the same relationship between its perimeters does not have to be replicated), to strengthen the meaning of each of these concepts. Trigonometry fits here, conceptually based on the similarity between triangles, an extremely useful topic in the measurement of numerous practical situations -in surveying, astronomy, satellite navigation, it can be used to calculate inaccessible distances through the measurements of angles and measurable sides of appropriate triangles. Furthermore, its subsequent amplification to the field of trigonometric functions constitutes a basic area for the modeling of physical phenomena, particularly electrical ones.

## THE ROLE OF CONTEXT

This type of teaching organization is metaphorically designated “helical”, which is the shape of a surface like that of a spiral staircase, since when climbing it you pass over a point already traveled, but with another height, representing the idea that over time the degree of formalization and complexity with which a previously discussed concept is worked on increases. A risk that is presented and we must try to avoid is that the helical strategy of approaching the issues becomes a “Carousel”, which always rotates and rotates at the same height level.

In accordance with the global context, references to information and communication technologies have been growing, transversally to the curriculum, as well as to interdisciplinarity. Regarding the latter, in particular in Santa Fe, there are proposals to address teaching through Interdisciplinary Content Nuclei in terms of bringing school disciplines into dialogue to promote the appropriation of socially significant knowledge by students. In effect, they suggest taking social problems, understood as events, as a starting point; among others: “problematic substance use”, “food”, “energy”, “the challenges of democracy”, “dengue”, “cultures”, “climate change”, “the universe”, “technology in the digital age”. This provides a concrete and conducive way to give voice to cultural and contextual issues to break them down from mathematics education in each singular classroom from situated institutional frameworks.

Likewise, in primary school classrooms, work on the Numbers and Operations axis is usually prioritized, and from a purely procedural level. This unbalances the other axes, such as Geometry and Measurement, which are sometimes addressed very superficially, leaving many relevant concepts or properties untreated.

Sometimes the teacher feels that he cannot “move on” to another topic because what is related to numbers and their operations has not yet been well learned by all the students. But you must keep in
geometric “objects” are also analyzed—parallelism and perpendicularity of lines; tangency between curves; adjacency of angles; and congruence and similarity of figures.

The Measurement part could be a separate axis since it has its own identity and sufficient relevance due to its direct applicability in numerous situations of daily life. Measurement systems are studied and students learn to measure time, angles, length, area, volume, capacity, both directly using some unit of measurement to measure objects; and indirectly, through the use of formulas and arithmetic operations. It is therefore an interesting space for integrating topics. For example, geometry with the field of numbers (to calculate the volume of a cylinder, two geometric elements must be identified that define its shape and size—the radius of the base r and the height h—and its measurements for the calculation Volume=$π*r^2*h$). The notion of independence of the measurements of perimeter, area and volume is worked on, analyzing concrete and simple situations of non-linearity (if the area of one figure is equal to twice the area of another, the same relationship between its perimeters does not have to be replicated), to strengthen the meaning of each of these concepts. Trigonometry fits here, conceptually based on the similarity between triangles, an extremely useful topic in the measurement of numerous practical situations -in surveying, astronomy, satellite navigation, it can be used to calculate inaccessible distances through the measurements of angles and measurable sides of appropriate triangles. Furthermore, its subsequent amplification to the field of trigonometric functions constitutes a basic area for the modeling of physical phenomena, particularly electrical ones.

## THE ROLE OF CONTEXT

This type of teaching organization is metaphorically designated “helical”, which is the shape of a surface like that of a spiral staircase, since when climbing it you pass over a point already traveled, but with another height, representing the idea that over time the degree of formalization and complexity with which a previously discussed concept is worked on increases. A risk that is presented and we must try to avoid is that the helical strategy of approaching the issues becomes a “carousel”, which always rotates and rotates at the same height level.

In accordance with the global context, references to information and communication technologies have been growing, transversally to the curriculum, as well as to interdisciplinarity. Regarding the latter, in particular in Santa Fe, there are proposals to address teaching through Interdisciplinary Content Nuclei in terms of bringing school disciplines into dialogue to promote the appropriation of socially significant knowledge by students. In effect, they suggest taking social problems, understood as events, as a starting point; among others: “problematic substance use”, “food”, “energy”, “the challenges of democracy”, “dengue”, “cultures”, “climate change”, “the universe”, “technology in the digital age”. This provides a concrete and conducive way to give voice to cultural and contextual issues to break them down from mathematics education in each singular classroom from situated institutional frameworks.

Likewise, in primary school classrooms, work on the Numbers and Operations axis is usually prioritized, and from a purely procedural level. This unbalances the other axes, such as Geometry and Measurement, which are sometimes addressed very superficially, leaving many relevant concepts or properties untreated.

Sometimes the teacher feels that he cannot “move on” to another topic because what is related to numbers and their operations has not yet been well learned by all the students. But you must keep in
mind that this topic can most likely be used in another subject through integration, such as in the calculation of volumes.

In the case of geometry specifically, many topics and aspects are not taught because they are unknown, due to the teacher's own school experiences. Typically, teachers have not been provided with appropriate formative understandings of this branch of mathematics. As a result, they generally view it with fear and distrust. Regarding measurement, those relating to volume and capacity are usually neglected, in line with the little or no treatment of the notion and calculation of volumes of bodies.

A similar phenomenon occurs in many Secondary Education classrooms, when the emphasis is on the simplification of long arithmetic or algebraic expressions, with combined operations, with multiple parentheses, brackets and keys, and strictly memorized cases of factoring, without prior motivations or later applications. The somewhat more formal approach to new topics in geometry such as congruence and similarity of figures and trigonometry seems to have no chance of being deployed. Among the arguments are those of the type "students need to master operations with algebraic expressions for higher studies", "they cannot understand the basics, how we are going to advance towards other things" and so on. These naturalized statements denote deep-rooted epistemological positions that condition and configure the teaching practices of mathematics.

However, at the prescribed level it is expected that all the axes in which school mathematics is organized will be developed. Furthermore, activities that integrate topics from different axes are promoted; for example, work with algebraic expressions can be combined with geometric interpretations, as well as with situations that can be modeled using such expressions. And even with historical-socio-cultural practices of the spatio-temporal context of the vast national territory, which remains a pending matter. In this framework, a student to be a teacher is trained in the specific aspects of the discipline/s that he/she will teach, placing himself at the educational level in which he will do so (Specific Training Field), learning the general humanistic and pedagogical foundations that will support his work as an educator (Field of General/Pedagogical Training) and having the possibility of deploying and rehearsing own abilities of teaching in making classroom decisions (Field of Professional Practice).

In the provincial proposal for the Teacher Training at the Initial Level of Education (2752 hours), the distribution of the minimum suggested hours for each Training Field is summarized as: Specific (1365, 50%), General (768 hours, 28%) and Practical Professional (619 hours, 22%). Punctually, mathematical disciplinary training is specified only in some portions of three curricular units, which also include aspects of didactics: Problem Solving and Creativity in the first year (42.7 h), Mathematics and its Didactics I in the second year (85.3 h), and Mathematics and its Didactics II in the third year (85.3 h). Thus, training in Mathematics is less than 213.3 h, which represents 16% of the Specific Training Field and, in turn, 8% of the total career proposal.

Likewise, within this scarcity, relevance is given to geometric issues, with two of the three blocks assigned to them: Number; Space and Geometry; Measurement. In addition, among the proposals for optional reduced formats, seminars on Ethnomathematics, geometry and movement, as well as plastic-visual language and Mathematics are included.
mind that this topic can most likely be used in another subject through integration, such as in the calculation of volumes.

In the case of geometry specifically, many topics and aspects are not taught because they are unknown, due to the teacher's own school experiences. Typically, teachers have not been provided with appropriate formative understandings of this branch of mathematics. As a result, they generally view it with fear and distrust. Regarding measurement, those relating to volume and capacity are usually neglected, in line with the little or no treatment of the notion and calculation of volumes of bodies.

A similar phenomenon occurs in many Secondary Education classrooms, when the emphasis is on the simplification of long arithmetic or algebraic expressions, with combined operations, with multiple parentheses, brackets and keys, and strictly memorized cases of factoring, without prior motivations or later applications. The somewhat more formal approach to new topics in geometry such as congruence and similarity of figures and trigonometry seems to have no chance of being deployed. Among the arguments are those of the type "students need to master operations with algebraic expressions for higher studies", "they cannot understand the basics, how we are going to advance towards other things" and so on. These naturalized statements denote deep-rooted epistemological positions that condition and configure the teaching practices of mathematics.

However, at the prescribed level it is expected that all the axes in which school mathematics is organized will be developed. Furthermore, activities that integrate topics from different axes are promoted; for example, work with algebraic expressions can be combined with geometric interpretations, as well as with situations that can be modeled using such expressions. And even with historical-socio-cultural practices of the spatio-temporal context of the vast national territory, which remains a pending matter. In this framework, a student to be a teacher is trained in the specific aspects of the discipline/s that he/she will teach, placing himself at the educational level in which he will do so (Specific Training Field), learning the general humanistic and pedagogical foundations that will support his work as an educator (Field of General/Pedagogical Training) and having the possibility of deploying and rehearsing own abilities of teaching in making classroom decisions (Field of Professional Practice).

In the provincial proposal for the Teacher Training at the Initial Level of Education (2752 hours), the distribution of the minimum suggested hours for each Training Field is summarized as: Specific (1365, 50%), General (768 hours, 28%) and Practical Professional (619 hours, 22%). Punctually, mathematical disciplinary training is specified only in some portions of three curricular units, which also include aspects of didactics: Problem Solving and Creativity in the first year (42.7 h), Mathematics and its Didactics I in the second year (85.3 h), and Mathematics and its Didactics II in the third year (85.3 h). Thus, training in Mathematics is less than 213.3 h, which represents 16% of the Specific Training Field and, in turn, 8% of the total career proposal.

Likewise, within this scarcity, relevance is given to geometric issues, with two of the three blocks assigned to them: Number; Space and Geometry; Measurement. In addition, among the proposals for optional reduced formats, seminars on Ethnomathematics, geometry and movement, as well as plastic-visual language and Mathematics are included.
In the Teacher Training at the Primary Level of Education (2816 h) the Specific Training Field is made up of the same spaces mentioned for the Initial Education Teaching Staff. The workloads by Field are similar: Specific (1429, 51%), General (768 hours, 27%) and Professional Practice (619 hours, 22%). Regarding training in Mathematics, an analogous proportion is allocated to the previous level (Initial), even slightly lower (15% of the Specific Training Field; 7.5% of the total Plan). How could a teacher, qualified to teach Mathematics many hours a week, from first to seventh grade (children from 6 to 12 years old), have acquired tools for effective professional development in such a short training time? In turn, the proposal for the area is arranged in six blocks, two of which are: Space and Geometry; Measurement.

In Argentina, the training of teachers to work in Secondary Education is already by discipline (mathematics in our case) and two types of institutions coexist: provincial higher education institutes and national universities. To provide context, currently in the field of state management, there are around 30 Mathematics Professorships (universities) and 14 at the provincial level in Santa Fe. The study plans that govern them and the institutional cultures are typical of each one (each university is autonomous and each province too). In these Teacher Training Courses, the Specific Training Fields comprise at least 60% of the total hourly load of the Study Plan. Among the branches that make up the mathematics area, there is a greater emphasis on calculus, with geometry along with algebra being the other two classic branches of development. Others mentioned are statistics and probability, mathematical modeling, discrete mathematics, and applications.

To this point, the relative weight of Geometry within the Curriculum is between 10 and 11%, and is developed gradually throughout the four years of the degree: Synthetic Geometry, Analytical Geometry, Dynamic Geometry, Euclidean geometry, Fractal geometry, Non-Euclidean geometries, Language in Geometry, Euclid's axiomatic system, Construction to conceptualization, Invariant properties, Transformations, Hilbert axiomatics, Contributions of Descartes and Klein, Erlangen Program, Incursion into Topology. In spaces of specific didactics and professional practice, disciplinary content is integrated with relevant theoretical currents and contexts located where future teachers carry out their immersion in real educational institutions. This is with the purpose of substantiating from initial training a specialized knowledge of the content (particular mathematical domain expected of a teacher). For example, calculating the perimeter of a rectangle requires activating knowledge different from that necessary to analyze a student's conjecture about a possible relationship between perimeter and area. The first requires only knowing ways to calculate the perimeter of the rectangle (common knowledge), while the second requires an ability to think flexibly about the notion of perimeter in order to analyze a statement, correct or not, of another person (knowledge specialized).

In general terms, it can be noted that, from the analysis of the curricular designs for teacher training in force, it emerges that in the case of Primary Education Teachers it is difficult to develop common knowledge of the content corresponding to that section, due to the scarcity of the time allocated to specific Mathematics topics (or geometry specifically). Hence the need for a review and readjustment, because a teacher cannot teach what he does not know, or even manage at the same level at which he must teach it. Even fewer will be able to articulate it in terms of specialized knowledge and/or integrate it with cultural practices in situated contexts.
In the Teacher Training at the Primary Level of Education (2816 h) the Specific Training Field is made up of the same spaces mentioned for the Initial Education Teaching Staff. The workloads by Field are similar: Specific (1429, 51%), General (768 hours, 27%) and Professional Practice (619 hours, 22%). Regarding training in Mathematics, an analogous proportion is allocated to the previous level (Initial), even slightly lower (15% of the Specific Training Field; 7.5% of the total Plan). How could a teacher, qualified to teach Mathematics many hours a week, from first to seventh grade (children from 6 to 12 years old), have acquired tools for effective professional development in such a short training time? In turn, the proposal for the area is arranged in six blocks, two of which are: Space and Geometry; Measurement.

In Argentina, the training of teachers to work in Secondary Education is already by discipline (mathematics in our case) and two types of institutions coexist: provincial higher education institutes and national universities. To provide context, currently in the field of state management, there are around 30 Mathematics Professorships (universities) and 14 at the provincial level in Santa Fe. The study plans that govern them and the institutional cultures are typical of each one (each university is autonomous and each province too). In these Teacher Training Courses, the Specific Training Fields comprise at least 60% of the total hourly load of the Study Plan. Among the branches that make up the mathematics area, there is a greater emphasis on calculus, with geometry along with algebra being the other two classic branches of development. Others mentioned are statistics and probability, mathematical modeling, discrete mathematics, and applications.

To this point, the relative weight of Geometry within the Curriculum is between 10 and 11%, and is developed gradually throughout the four years of the degree: Synthetic Geometry, Analytical Geometry, Dynamic Geometry, Euclidean geometry, Fractal geometry, Non-Euclidean geometries, Language in Geometry, Euclid's axiomatic system, Construction to conceptualization, Invariant properties, Transformations, Hilbert axiomatics, Contributions of Descartes and Klein, Erlangen Program, Incursion into Topology. In spaces of specific didactics and professional practice, disciplinary content is integrated with relevant theoretical currents and contexts located where future teachers carry out their immersion in real educational institutions. This is with the purpose of substantiating from initial training a specialized knowledge of the content (particular mathematical domain expected of a teacher). For example, calculating the perimeter of a rectangle requires activating knowledge different from that necessary to analyze a student's conjecture about a possible relationship between perimeter and area. The first requires only knowing ways to calculate the perimeter of the rectangle (common knowledge), while the second requires an ability to think flexibly about the notion of perimeter in order to analyze a statement, correct or not, of another person (knowledge specialized).

In general terms, it can be noted that, from the analysis of the curricular designs for teacher training in force, it emerges that in the case of Primary Education Teachers it is difficult to develop common knowledge of the content corresponding to that section, due to the scarcity of the time allocated to specific Mathematics topics (or geometry specifically). Hence the need for a review and readjustment, because a teacher cannot teach what he does not know, or even manage at the same level at which he must teach it. Even fewer will be able to articulate it in terms of specialized knowledge and/or integrate it with cultural practices in situated contexts.
DISCUSSION

In its origins, the province of Santa Fe was a territory inhabited by groups of natives, organized into tribes (Tobas, Guaraníes, Mocovíes, Pilagás, Guaycurúes, Querandíes, Abipones, Timbúes, Quiloazas, Mocoretás, and Corondas). Inquiring into their worldviews, in connection with nature and about how geometry has been intrinsically present in their practices also presents fertile, unexplored territory. As an example, cultural diversity is celebrated on October 12 each year—it is a very emotional moment of reflection that can be recovered through school projects that bring disciplines, or branches of these, into dialogue, such as geometry. Some possibilities may be to explore geometric shapes in cultural legends, develop geometric art inspired by cultures, study monuments and iconic structures of various cultures, recognize the presence of geometry in Nature, or build geometric models.

At the national and provincial level, in particular, agricultural and industrial-based activity is influential, proving interesting activities that can support geometry with transversal themes such as environment, gender, technologies, and human rights. This is on the agenda, with initial developments in the daily classroom. Starting with Teacher Training, initial and continuous, is considered a favorable path for this; punctually, from socialization of examples of similar practices in relatively close fields.
DISCUSSION

In its origins, the province of Santa Fe was a territory inhabited by groups of natives, organized into tribes (Tobas, Guaraníes, Mocovíes, Pilagás, Guaycurúes, Querandíes, Abipones, Timbúes, Quiloazas, Mocoretás, and Corondas). Inquiring into their worldviews, in connection with nature and about how geometry has been intrinsically present in their practices also presents fertile, unexplored territory. As an example, cultural diversity is celebrated on October 12 each year—it is a very emotional moment of reflection that can be recovered through school projects that bring disciplines, or branches of these, into dialogue, such as geometry. Some possibilities may be to explore geometric shapes in cultural legends, develop geometric art inspired by cultures, study monuments and iconic structures of various cultures, recognize the presence of geometry in Nature, or build geometric models.

At the national and provincial level, in particular, agricultural and industrial-based activity is influential, proving interesting activities that can support geometry with transversal themes such as environment, gender, technologies, and human rights. This is on the agenda, with initial developments in the daily classroom. Starting with Teacher Training, initial and continuous, is considered a favorable path for this; punctually, from socialization of examples of similar practices in relatively close fields.
<figure><img src="image_7.png" /></figure>
<figure><img src="image_6.png" /></figure>
# A GLIMPSE OF THE PRACTICES OF GEOMETRY EDUCATION IN IRAN

Zahra Gooya

Shahid Beheshti University

## INTRODUCTION

The purpose of this paper is to provide a glimpse of the evolution of geometry education from the foundation of the first modern education institution in the late 18th century to the establishment of the formal education system in 1925. From inception until 2011, geometry education was divided into three different periods. This paper will give a brief account of the characteristics of each period in terms of the geometry curricula/textbooks in Iran.

Geometry is part of the Iranian scientific history and the contribution of Iranian mathematicians to the advancement of geometry has been remarkable in the history of mathematics. Scholars include Abu al-wafa Buzjani (10th century), al-Karaji (11th century), Khayyam (11th to 12th century) and Nasir al-Din al-Tusi (12th century), to name a few. Furthermore, geometry has been part of the Iranian culture, artistry, architecture, and handcrafts. Nevertheless, geometry education in Iran, did not take the soul of this cultural wealth and in contrast, was influenced by abstract aspects of geometry adapted from French geometry curricula. In the following, I will take a glimpse at the geometry education from Dar ul-Funun<fnref n="1" /> to the new century.

### Geometry Education since the first modern institution

Dar ul-Funun was the first modern education institution in Iran, founded in Tehran in 1851. The tradition of geometry education in Dar ul-Funun was drawn from French traditions. Teachers taught in their own language and translators were in classes to translate them into Farsi for students. Gradually several of the translators started to write geometry textbooks in Farsi with an emphasis on abstraction and rigor as presented in Euclid's Elements. This synthetic approach had almost no relation with traditional geometry that was developed in Iran in the past.

After the establishment of formal secondary education in Iran in 1925, the senior high school started with two streams: namely, “science” and “literature”. Geometry was one of the main subjects of the science stream. The approach to geometry education continued the tradition of Dar ul-Funun and focused on deductive reasoning and proof. However, in the beginning of formal secondary education, the number of students was low and those choosing the science stream for the senior high school were limited. Typically, elite students entered the science stream and for most of them, the abstract level of deductive practice of geometry was attractive and manageable. With the expansion of secondary education, this approach prevented many students selecting the “science stream” and the abstract approach to mathematics education, acted as “gatekeeper” for general students.

From the creation of the secondary education in 1925, three distinct periods of practice took place.

<footnote n="1">¹ polytechnic</footnote>
# A GLIMPSE OF THE PRACTICES OF GEOMETRY EDUCATION IN IRAN

Zahra Gooya

Shahid Beheshti University

## INTRODUCTION

The purpose of this paper is to provide a glimpse of the evolution of geometry education from the foundation of the first modern education institution in the late 18th century to the establishment of the formal education system in 1925. From inception until 2011, geometry education was divided into three different periods. This paper will give a brief account of the characteristics of each period in terms of the geometry curricula/textbooks in Iran.

Geometry is part of the Iranian scientific history and the contribution of Iranian mathematicians to the advancement of geometry has been remarkable in the history of mathematics. Scholars include Abu al-wafa Buzjani (10th century), al-Karaji (11th century), Khayyam (11th to 12th century) and Nasir al-Din al-Tusi (12th century), to name a few. Furthermore, geometry has been part of the Iranian culture, artistry, architecture, and handcrafts. Nevertheless, geometry education in Iran, did not take the soul of this cultural wealth and in contrast, was influenced by abstract aspects of geometry adapted from French geometry curricula. In the following, I will take a glimpse at the geometry education from Dar ul-Funun<fnref n="1" /> to the new century.

### Geometry Education since the first modern institution

Dar ul-Funun was the first modern education institution in Iran, founded in Tehran in 1851. The tradition of geometry education in Dar ul-Funun was drawn from French traditions. Teachers taught in their own language and translators were in classes to translate them into Farsi for students. Gradually several of the translators started to write geometry textbooks in Farsi with an emphasis on abstraction and rigor as presented in Euclid's Elements. This synthetic approach had almost no relation with traditional geometry that was developed in Iran in the past.

After the establishment of formal secondary education in Iran in 1925, the senior high school started with two streams: namely, “science” and “literature”. Geometry was one of the main subjects of the science stream. The approach to geometry education continued the tradition of Dar ul-Funun and focused on deductive reasoning and proof. However, in the beginning of formal secondary education, the number of students was low and those choosing the science stream for the senior high school were limited. Typically, elite students entered the science stream and for most of them, the abstract level of deductive practice of geometry was attractive and manageable. With the expansion of secondary education, this approach prevented many students selecting the “science stream” and the abstract approach to mathematics education, acted as “gatekeeper” for general students.

From the creation of the secondary education in 1925, three distinct periods of practice took place.

<footnote n="1">¹ polytechnic</footnote>
**First period from 1942 to 1970:** In this period, the Ministry of Education allowed different publishers to produce textbooks for each subject from different perspectives, including geometry. However, the approach to geometry education continued to focus on deductive reasoning and proof. In this period, various changes took place at the senior secondary science stream and later, the "mathematics-physics" stream, but geometry subjects autonomously. There were various school subjects under the geometry umbrella such as "descriptive geometry", "spatial geometry", "analytical geometry" and "conic sections", all faithful to an axiomatic, Euclidian approach.

**Second period from 1970 to 1995:** In early 1970's the school mathematics curriculum in Iran adapted the "new math", except for geometry education and geometry textbooks. Geometry endured intact, despite Dieudonné's desire to replace Euclidean approaches with transformational geometry based on vectors and vector space in two dimensions (Dieudonné, 1972). In fact, Euclidian approaches were mandated until 1995 despite not being aligned with the traditional Iranian geometry. European influences prevailed with the expansion of formal education and the rapid increase of high school students. Geometry textbooks were written to teach proof, using deductive reasoning.

After the revolution of 1979 and the end of the eight-year Iran-Iraq war in 1988, the secondary education system underwent change in 1991 and the curriculum and textbooks for all subjects changed accordingly.2 Regarding mathematics curriculum, the “new math” subject including the emphasis on formal logic was eliminated and new textbooks were written towards moderate reform and as usual, two geometry textbooks for grades nine and ten, were written with the same axiomatic Euclidean tradition. However, in contrast to the past, the number of students entering secondary education and willing to choose mathematics-physics and science streams increased rapidly ensuring geometry’s audience was much broader. The consequence was that in less than one year, and after the report of the national assessment of progress in geometry (Kiamanesh, 1994), the Ministry decided to re-author two geometry textbooks. I was invited with several others to develop the geometry curriculum and write Geometry I and Geometry II textbooks3. From my perspective, this was an opportunity to take a different stance on geometry education, allowing secondary students to enjoy learning geometry. To start the new endeavor, we first reviewed the historical development of geometry education from Dar ul-Funun to two periods of the foundation of formal education system in Iran and the influence of new math describing the third period.

**Third period from 1995 to 2014:** To begin the work, we believed that cultural practices of geometry in Iran, its tradition and its place in the Iranian heritage should be our first step towards developing the curriculum. As well, the historical trend of modern and formal education should be considered seriously. In the meanwhile, I participated at PME 18 in Lisbon in 1994 when the study document for the 14th ICMI Study titled "Perspectives on the teaching of geometry for the 21st century" was released. By reviewing the related documents at the national and international level, we concluded that school geometry should encourage students to formulate conjectures and proofs, through activities that were accessible to as many students as possible. Relations between intuition, inductive

<footnote n="2">² Iran education system is centralized and there is one national textbook for every school subject.</footnote>

<footnote n="3">³ The details of this development have been presented in various conferences and the team and myself, have written several papers to describe that new trend. The last paper written by Gholamazad, Gooya & Zangeneh is ready for submission.</footnote>
**First period from 1942 to 1970:** In this period, the Ministry of Education allowed different publishers to produce textbooks for each subject from different perspectives, including geometry. However, the approach to geometry education continued to focus on deductive reasoning and proof. In this period, various changes took place at the senior secondary science stream and later, the "mathematics-physics" stream, but geometry subjects autonomously. There were various school subjects under the geometry umbrella such as "descriptive geometry", "spatial geometry", "analytical geometry" and "conic sections", all faithful to an axiomatic, Euclidian approach.

**Second period from 1970 to 1995:** In early 1970's the school mathematics curriculum in Iran adapted the "new math", except for geometry education and geometry textbooks. Geometry endured intact, despite Dieudonné's desire to replace Euclidean approaches with transformational geometry based on vectors and vector space in two dimensions (Dieudonné, 1972). In fact, Euclidian approaches were mandated until 1995 despite not being aligned with the traditional Iranian geometry. European influences prevailed with the expansion of formal education and the rapid increase of high school students. Geometry textbooks were written to teach proof, using deductive reasoning.

After the revolution of 1979 and the end of the eight-year Iran-Iraq war in 1988, the secondary education system underwent change in 1991 and the curriculum and textbooks for all subjects changed accordingly.2 Regarding mathematics curriculum, the “new math” subject including the emphasis on formal logic was eliminated and new textbooks were written towards moderate reform and as usual, two geometry textbooks for grades nine and ten, were written with the same axiomatic Euclidean tradition. However, in contrast to the past, the number of students entering secondary education and willing to choose mathematics-physics and science streams increased rapidly ensuring geometry’s audience was much broader. The consequence was that in less than one year, and after the report of the national assessment of progress in geometry (Kiamanesh, 1994), the Ministry decided to re-author two geometry textbooks. I was invited with several others to develop the geometry curriculum and write Geometry I and Geometry II textbooks3. From my perspective, this was an opportunity to take a different stance on geometry education, allowing secondary students to enjoy learning geometry. To start the new endeavor, we first reviewed the historical development of geometry education from Dar ul-Funun to two periods of the foundation of formal education system in Iran and the influence of new math describing the third period.

**Third period from 1995 to 2014:** To begin the work, we believed that cultural practices of geometry in Iran, its tradition and its place in the Iranian heritage should be our first step towards developing the curriculum. As well, the historical trend of modern and formal education should be considered seriously. In the meanwhile, I participated at PME 18 in Lisbon in 1994 when the study document for the 14th ICMI Study titled "Perspectives on the teaching of geometry for the 21st century" was released. By reviewing the related documents at the national and international level, we concluded that school geometry should encourage students to formulate conjectures and proofs, through activities that were accessible to as many students as possible. Relations between intuition, inductive

<footnote n="2">² Iran education system is centralized and there is one national textbook for every school subject.</footnote>

<footnote n="3">³ The details of this development have been presented in various conferences and the team and myself, have written several papers to describe that new trend. The last paper written by Gholamazad, Gooya & Zangeneh is ready for submission.</footnote>
and deductive proofs, age of students at which proofs can be introduced, and different levels of rigor and abstraction needed to be accounted for.

Geometry I started with simple and computational/algebraic examples and used axioms implicitly that were based on student experiences. For example, for the similarity of two triangles, we relied on what students knew from junior high school related to the concept of rectangle and area rather than focusing on computational examples. We used inductive and deductive reasoning across the book. Students calculated the volume of spheres by first calculating the area of circles.

Geometry 2 encouraged students to use inequality, deductive reasoning, and indirect proof as tools and methods for direct proof. Analytic approaches included considering transformational geometry with coordinated geometry. A serious difficulty for university graduates was that they could not use their higher mathematics knowledge fluently to teach axiomatic Euclidian high school geometry. However, the new approach gave them confidence to use their new knowledge for teaching geometry. For the development of Geometry I and Geometry II textbooks, the following principles were considered:

1. Guessing (conjecturing) using inductive reasoning and proving using mathematical induction is a great opportunity for students to develop logical thinking.

2. Coordination and unification of different parts of mathematics as a legacy of Golden Age of Iranian-Islamic era and its potential that was the opposite of new math in which various mathematics branches, topics or even concepts dealt with differently.

3. Teaching theorems are important, but not in a way that in traditional or new math approach, teaching a theorem and its proof with all its details at once. On the contrary, we broke famous theorems such as Thales into several parts and through suitable activities, students engaged in doing the proof with other students in small groups and via whole class discussions and contributed to the development of proof. In this manner, each part acted as a scaffolder to build the other part and at last, the textbook brought all pieces in one big picture as "proving the theorem".

To give an example, the following activity is taken from Geometry 1 (1995):

*This regular dodecahedron has been divided into six parts. Cut the pieces and put them together in a way that makes a square. Note that one of the pieces is an equilateral triangle. In the first step, students should solve it by manipulating the pieces. But in the next step they were asked to show it is a square. In other words they should prove that the new figure is a square.*

<figure><img src="image_8.png" /><figcaption>Figure 1. Puzzle posed by Abu al-Wafa al-Buzjani (11th century)</figcaption></figure>
and deductive proofs, age of students at which proofs can be introduced, and different levels of rigor and abstraction needed to be accounted for.

Geometry I started with simple and computational/algebraic examples and used axioms implicitly that were based on student experiences. For example, for the similarity of two triangles, we relied on what students knew from junior high school related to the concept of rectangle and area rather than focusing on computational examples. We used inductive and deductive reasoning across the book. Students calculated the volume of spheres by first calculating the area of circles.

Geometry 2 encouraged students to use inequality, deductive reasoning, and indirect proof as tools and methods for direct proof. Analytic approaches included considering transformational geometry with coordinated geometry. A serious difficulty for university graduates was that they could not use their higher mathematics knowledge fluently to teach axiomatic Euclidian high school geometry. However, the new approach gave them confidence to use their new knowledge for teaching geometry. For the development of Geometry I and Geometry II textbooks, the following principles were considered:

1. Guessing (conjecturing) using inductive reasoning and proving using mathematical induction is a great opportunity for students to develop logical thinking.

2. Coordination and unification of different parts of mathematics as a legacy of Golden Age of Iranian-Islamic era and its potential that was the opposite of new math in which various mathematics branches, topics or even concepts dealt with differently.

3. Teaching theorems are important, but not in a way that in traditional or new math approach, teaching a theorem and its proof with all its details at once. On the contrary, we broke famous theorems such as Thales into several parts and through suitable activities, students engaged in doing the proof with other students in small groups and via whole class discussions and contributed to the development of proof. In this manner, each part acted as a scaffolder to build the other part and at last, the textbook brought all pieces in one big picture as "proving the theorem".

To give an example, the following activity is taken from Geometry 1 (1995):

*This regular dodecahedron has been divided into six parts. Cut the pieces and put them together in a way that makes a square. Note that one of the pieces is an equilateral triangle. In the first step, students should solve it by manipulating the pieces. But in the next step they were asked to show it is a square. In other words they should prove that the new figure is a square.*

<figure><img src="image_7.png" /><figcaption>Figure 1. Puzzle posed by Abu al-Wafa al-Buzjani (11th century)</figcaption></figure>
## Discussion

Geometry has strongly rooted in Iranian culture and thus, in every period of schooling, has had its own dominant place in school mathematics curricula. Since the foundation of Dar ul-Funun and later, the establishment of formal education system in Iran, geometry has always been “of its own merit” and not necessarily regarded as part of school mathematics. Reviewing the history of geometry in Iran, shows that great mathematicians, philosophers, and poets created the most sophisticated geometry that have boosted mathematics and science throughout the history of humankind and had a certain role in the development of the western renaissance. This cultural and traditional geometry is heavily based on practices and real-life problems and its move has been from intuition and reality to abstraction. Geometry is strongly rooted in Iranian culture, and it is the soul and base of many artifacts, buildings, handcrafts, and carpet weavings and is part of our cultural heritage. However, since the modern and formal education was imported from Europe and particularly France, the approach to its education as school subject, was purely focused on the Euclidean tradition. This resulted in two independent and strong culture practices in geometry education in Iran. Before the reform of 1995, the formal geometry education in Iran was disconnected from the real world, and the other mathematics subjects as well. Its Euclidean axiomatic approach, prepared the scene for memorization of facts, proofs and procedures, and the language used was unfamiliar for students.

## CONCLUSION

However, the beauty and flexibility of geometry is partly due to its capacity for developing intuition, visualization, imagination, and application in the real world that is not limited to an axiomatic framework. In our view, this is the central reason for geometry having a self-contained place in school mathematics since no other parts of mathematics have the potential of geometry to include both extremes of far intuition and far abstraction within one frame! In addition, geometry is closely connected with culture and cultural artifacts and people in different regions around the world feel that they are part of this endeavor and thus could contribute to its development. Rather than using artificial cultural symbols, it is more appropriate to implement the findings of various ethnomathematics to help students to see the relation between mathematics and culture (Gooya & Gholamazad, 2021).

## References

- Dieudonné, J. (1972). Panel Discussion of Research in Mathematics. Second Iranian International Conference. Aryamehr (Sharif) University of Technology. Tehran. Iran.

- Gooya, Z., & Gholamazad, S. (2021). The Covid-19 pandemic: A challenge for values in school mathematics in Iran. JERM, 31(3), 379–392. (In Farsi.)

- Kiamanesh, A. (1994). Assessment of the mathematics and geometry courses of the new secondary education system. Quarterly Journal of Education. Fall & Winter. No. 39 & 40, 25-40. Ministry of Education. (In Farsi.)
**Discussion**

Geometry has strongly rooted in Iranian culture and thus, in every period of schooling, has had its own dominant place in school mathematics curricula. Since the foundation of Dar ul-Funun and later, the establishment of formal education system in Iran, geometry has always been “of its own merit” and not necessarily regarded as part of school mathematics. Reviewing the history of geometry in Iran, shows that great mathematicians, philosophers, and poets created the most sophisticated geometry that have boosted mathematics and science throughout the history of humankind and had a certain role in the development of the western renaissance. This cultural and traditional geometry is heavily based on practices and real-life problems and its move has been from intuition and reality to abstraction. Geometry is strongly rooted in Iranian culture, and it is the soul and base of many artifacts, buildings, handcrafts, and carpet weavings and is part of our cultural heritage. However, since the modern and formal education was imported from Europe and particularly France, the approach to its education as school subject, was purely focused on the Euclidean tradition. This resulted in two independent and strong culture practices in geometry education in Iran. Before the reform of 1995, the formal geometry education in Iran was disconnected from the real world, and the other mathematics subjects as well. Its Euclidean axiomatic approach, prepared the scene for memorization of facts, proofs and procedures, and the language used was unfamiliar for students.

**CONCLUSION**

However, the beauty and flexibility of geometry is partly due to its capacity for developing intuition, visualization, imagination, and application in the real world that is not limited to an axiomatic framework. In our view, this is the central reason for geometry having a self-contained place in school mathematics since no other parts of mathematics have the potential of geometry to include both extremes of far intuition and far abstraction within one frame! In addition, geometry is closely connected with culture and cultural artifacts and people in different regions around the world feel that they are part of this endeavor and thus could contribute to its development. Rather than using artificial cultural symbols, it is more appropriate to implement the findings of various ethnomathematics to help students to see the relation between mathematics and culture (Gooya & Gholamazad, 2021).

**References**

Dieudonné, J. (1972). Panel Discussion of Research in Mathematics. Second Iranian International Conference. Aryamehr (Sharif) University of Technology. Tehran. Iran.

Gooya, Z., & Gholamazad, S. (2021). The Covid-19 pandemic: A challenge for values in school mathematics in Iran. *JERM, 31*(3), 379–392. (In Farsi.)

Kiamanesh, A. (1994). Assessment of the mathematics and geometry courses of the new secondary education system. Quarterly Journal of Education. Fall & Winter. No. 39 & 40, 25-40. Ministry of Education. (In Farsi.)
# ETHNOMODELS OF LANDLESS PEOPLES’ MOVEMENT: MEASURING LAND – BRAZIL

Milton Rosa

Universidade Federal de Ouro Preto

## INTRODUCTION

In the struggle for agrarian reform, the access to a plot of land and to live and produce on it makes the practices of measuring the land to be a central activity of the members of the *Landless Peoples' Movement (Movimento dos Sem Terra - MST)*, in Southern Brazil, mainly because of the importance placed on sustainability and planning of agricultural production. In the study conducted by Knijnik (1993) it was proposed that the elaboration of curricular mathematics activities related to the demarcation of land with participants of this movement. These activities were related to the method of *cubação of land*, which is a traditional mathematical practice applied by participants of this specific cultural group to measure and determine the area of the land in their settlements (occupation sites).

The daily necessities of the MST members caused them to capture the procedures of these techniques, showing that, despite their low level of schooling, they were able to apprehend and apply knowledge related to the methods of cubação of land, which is one of the tools used to solve problems related to the measurement of land with irregular shapes by applying distinct methods to determine this area. According to Knijnik (1996), this method met the specific needs of the members of this movement because they applied it to determine land areas related to the delimitation of planting sectors as well to demarcate the plot of land of each family in the settlement.

They established production goals related to their own logistic possibilities such as storage and drying, bagging, transport, and selling products at local markets. The land worked by MST members was prepared according to the type of farm and quantity of the product these members harvested and commercialized. The emic knowledge related to the development of these methods was orally transmitted and diffused to MST family members by their ancestors across generations. Thus, the mathematical knowledge involved in these local methods is also related to productive activities that members of this specific cultural group performed in their daily routines. For example, the need for the development of *cubação of land* with irregular shapes was in accordance with its accessibility depending on its topology and the quality of desired agricultural products.

This method is used to calculate the total area of a land in order to calculate the amount of money needed to be paid or received for the cleaning work of the property, or for the preparation of the land for planting as well the demarcation of areas to be cultivated, to plan and to delimitate areas for the construction of houses and shelters for animals. It is also used to make payments for work done by the members of a settlement in the state of Bahia, in Brazil, according to the land *frames* or shapes. For example, there is job related to land with *two corners*, *three corners*, or *four corners* in accordance with the shape of the cultivated area (Silva, 2012). According to D'Ambrosio (1999), the validation of these methods within agricultural communities and settlements results from the development of
# ETHNOMODELS OF LANDLESS PEOPLES’ MOVEMENT: MEASURING LAND – BRAZIL

Milton Rosa

Universidade Federal de Ouro Preto

## INTRODUCTION

In the struggle for agrarian reform, the access to a plot of land and to live and produce on it makes the practices of measuring the land to be a central activity of the members of the *Landless Peoples' Movement (Movimento dos Sem Terra - MST)*, in Southern Brazil, mainly because of the importance placed on sustainability and planning of agricultural production. In the study conducted by Knijnik (1993) it was proposed that the elaboration of curricular mathematics activities related to the demarcation of land with participants of this movement. These activities were related to the method of *cubação of land*, which is a traditional mathematical practice applied by participants of this specific cultural group to measure and determine the area of the land in their settlements (occupation sites).

The daily necessities of the MST members caused them to capture the procedures of these techniques, showing that, despite their low level of schooling, they were able to apprehend and apply knowledge related to the methods of cubação of land, which is one of the tools used to solve problems related to the measurement of land with irregular shapes by applying distinct methods to determine this area. According to Knijnik (1996), this method met the specific needs of the members of this movement because they applied it to determine land areas related to the delimitation of planting sectors as well to demarcate the plot of land of each family in the settlement.

They established production goals related to their own logistic possibilities such as storage and drying, bagging, transport, and selling products at local markets. The land worked by MST members was prepared according to the type of farm and quantity of the product these members harvested and commercialized. The emic knowledge related to the development of these methods was orally transmitted and diffused to MST family members by their ancestors across generations. Thus, the mathematical knowledge involved in these local methods is also related to productive activities that members of this specific cultural group performed in their daily routines. For example, the need for the development of *cubação of land* with irregular shapes was in accordance with its accessibility depending on its topology and the quality of desired agricultural products.

This method is used to calculate the total area of a land in order to calculate the amount of money needed to be paid or received for the cleaning work of the property, or for the preparation of the land for planting as well the demarcation of areas to be cultivated, to plan and to delimitate areas for the construction of houses and shelters for animals. It is also used to make payments for work done by the members of a settlement in the state of Bahia, in Brazil, according to the land *frames* or shapes. For example, there is job related to land with *two corners*, *three corners*, or *four corners* in accordance with the shape of the cultivated area (Silva, 2012). According to D'Ambrosio (1999), the validation of these methods within agricultural communities and settlements results from the development of
informal agreements of signification that results from a long cumulative process of generation, intellectual organization, social organization, and diffusion of this knowledge.

# CUBAÇÃO WITH QUADRILATERAL SHAPES: CALCULATION OF LAND AREA WITH FOUR CORNERS

Mathematical practices investigated in the study conducted by Knijnik (1993) consisted of two methods that were called by her students in the classroom as *Adão's Method* and *Jorge's Method*. These two students who were members of the *Landless Peoples' Movement (MST)* presented, explained, and taught these specific ways of measuring their land in their settlements to the other learners in the classroom. The investigation of these two methods discusses the interrelations between local (emic) and academic (etic) mathematical knowledge concerning the upper bound estimation of the area of a tract of land with irregular shapes.

## Transforming the Shape of an Irregular Quadrilateral into a Rectangle

The first method is called the *Adão's Method* that transforms the shape of an irregular quadrilateral into a rectangle. In this context, Adão explained how to determine his method, which we consider as an emic ethnomodel:

Well folks, this is the most common formula that is used on the countryside, up there on the farm, right? And, let's assume that I am the owner of a crop and I lent this *frame* here to a friend *to mow* and I told him that I will pay three thousand by the *fourth*. Then, he mowed this land and he even *passed the rope* himself to find its area. Then, he measured this *wall* here, 90 meters, the other, 152 meters, 114 meters, 124 meters. Did you notice that there is no *wall*, no base, and no height that has the same measure, right? The two landmarks that are *lying down* are the *bases* and the *heights* are those that are *standing up*. Ok. So, I did the following here, right: I added the two bases and divided the sum by 2. I found 138. So, the base is 138 here and 138 there, understood? So, I have here the two heights, 114 plus 90. I found 204 and divided it by 2, 102, right? So, now we just need to multiply the base times height, Ok? I think the answer is 14076 square meters, right? This is the area that he *mowed*.

<figure><img src="image_9.png" /></figure>

It is important to state that, during his narrative, Adão used expressions such as:

a) *Walls* (paredes) that mean the landmarks of the land.

b) *Frame* (quadro) that means the area of a land with a quadrilateral shape.

c) *To mow* (carpir) means to clean or to prepare the land for planting.

d) *Fourth* (quarta) that means an area measurement used in the Brazilian rural context that is equivalent to a quarter of a Paulista bushel that is used in the state of São Paulo, Brazil, which measures 24200 square meters.

e) *Pass the rope* (passar a corda) means to measure the land by using a rope.
informal agreements of signification that results from a long cumulative process of generation, intellectual organization, social organization, and diffusion of this knowledge.

# CUBAÇÃO WITH QUADRILATERAL SHAPES: CALCULATION OF LAND AREA WITH FOUR CORNERS

Mathematical practices investigated in the study conducted by Knijnik (1993) consisted of two methods that were called by her students in the classroom as *Adão's Method* and *Jorge's Method*. These two students who were members of the *Landless Peoples' Movement (MST)* presented, explained, and taught these specific ways of measuring their land in their settlements to the other learners in the classroom. The investigation of these two methods discusses the interrelations between local (emic) and academic (etic) mathematical knowledge concerning the upper bound estimation of the area of a tract of land with irregular shapes.

## Transforming the Shape of an Irregular Quadrilateral into a Rectangle

The first method is called the *Adão's Method* that transforms the shape of an irregular quadrilateral into a rectangle. In this context, Adão explained how to determine his method, which we consider as an emic ethnomodel:

Well folks, this is the most common formula that is used on the countryside, up there on the farm, right? And, let's assume that I am the owner of a crop and I lent this *frame* here to a friend *to mow* and I told him that I will pay three thousand by the *fourth*. Then, he mowed this land and he even *passed the rope* himself to find its area. Then, he measured this *wall* here, 90 meters, the other, 152 meters, 114 meters, 124 meters. Did you notice that there is no *wall*, no base, and no height that has the same measure, right? The two landmarks that are *lying down* are the *bases* and the *heights* are those that are *standing up*. Ok. So, I did the following here, right: I added the two bases and divided the sum by 2. I found 138. So, the base is 138 here and 138 there, understood? So, I have here the two heights, 114 plus 90. I found 204 and divided it by 2, 102, right? So, now we just need to multiply the base times height, Ok? I think the answer is 14076 square meters, right? This is the area that he *mowed*.

<figure><img src="image_8.png" /></figure>

It is important to state that, during his narrative, Adão used expressions such as:

a) *Walls* (paredes) that mean the landmarks of the land.

b) *Frame* (quadro) that means the area of a land with a quadrilateral shape.

c) *To mow* (carpir) means to clean or to prepare the land for planting.

d) *Fourth* (quarta) that means an area measurement used in the Brazilian rural context that is equivalent to a quarter of a Paulista bushel that is used in the state of São Paulo, Brazil, which measures 24200 square meters.

e) *Pass the rope* (passar a corda) means to measure the land by using a rope.
These terms are the vocabularies (jargons) used by the members of this distinct cultural group to describe the procedures of the development of their local mathematical practices. Table 1 shows Adão's method of estimating an area of a land with irregular shape.

<table><caption>Table 1: Adão's method of estimating an area of a land with irregular shape. Source: Adapted from Knijnik (1993, p. 24)</caption><thead><tr><td>Adão's explanation (Emic/Local knowledge)</td><td>Academic explanation (Etic/Global Knowledge)</td></tr></thead><tbody><tr><td>This is a piece of land with four walls</td><td>This is a convex quadrilateral</td></tr><tr><td>First, we add two of the opposite walls and divide them by two</td><td>First, we find the average of two opposite sides</td></tr><tr><td>Second, we add the other two opposite sides and also divide them by two</td><td>Second, we find the average of the other two opposite sides</td></tr><tr><td>Third, we multiply the first obtained number by the second one</td><td>Third, we determine the product of the two average numbers previously determined</td></tr><tr><td>That is the cubação of the land</td><td>This is the area of the rectangle whose sides are the average of the two pairs of opposite sides of the convex quadrilateral</td></tr></tbody></table>

Thus, this emic (local) mathematical knowledge can be represented by an etic (global) ethnomodel that transforms the shape of the given land into a rectangle of 138 metres x 102 metres with an area of 14076 square meters.

$$ 
\begin{aligned}
\text{Area} &= \left(\frac{a+c}{2}\right) \times \left(\frac{b+d}{2}\right) \\
\text{Area} &= \left(\frac{124+152}{2}\right) + \left(\frac{90+114}{2}\right) \\
\text{Area} &= \left(\frac{276}{2}\right) \times \left(\frac{204}{2}\right) \\
\text{Area} &= 138 \times 102
\end{aligned}
$$

$$ \text{Area} = 14076 \text{ square meters} $$

The representation of this mathematical practice can be explained by the following etic ethnomodel procedures: a) transform the shape of the irregular quadrilateral in a rectangle whose area can be determined through the application of the area formula, b) determine the dimensions of the rectangle by calculating the average of the two opposite sides of the irregular quadrilateral, and c) determine the area of the rectangle by applying the formula: $A = b \times h$.

<figure><img src="image_10.png" /></figure>
These terms are the vocabularies (jargons) used by the members of this distinct cultural group to describe the procedures of the development of their local mathematical practices. Table 1 shows Adão's method of estimating an area of a land with irregular shape.

<table><caption>Table 1: Adão's method of estimating an area of a land with irregular shape. Source: Adapted from Knijnik (1993, p. 24)</caption><thead><tr><td>Adão's explanation (Emic/Local knowledge)</td><td>Academic explanation (Etic/Global Knowledge)</td></tr></thead><tbody><tr><td>This is a piece of land with four walls</td><td>This is a convex quadrilateral</td></tr><tr><td>First, we add two of the opposite walls and divide them by two</td><td>First, we find the average of two opposite sides</td></tr><tr><td>Second, we add the other two opposite sides and also divide them by two</td><td>Second, we find the average of the other two opposite sides</td></tr><tr><td>Third, we multiply the first obtained number by the second one</td><td>Third, we determine the product of the two average numbers previously determined</td></tr><tr><td>That is the cubação of the land</td><td>This is the area of the rectangle whose sides are the average of the two pairs of opposite sides of the convex quadrilateral</td></tr></tbody></table>

Thus, this emic (local) mathematical knowledge can be represented by an etic (global) ethnomodel that transforms the shape of the given land into a rectangle of 138 metres x 102 metres with an area of 14076 square meters.

$$ 
\begin{aligned}
\text{Area} &= \left(\frac{a+c}{2}\right) \times \left(\frac{b+d}{2}\right) \\
\text{Area} &= \left(\frac{124+152}{2}\right) + \left(\frac{90+114}{2}\right) \\
\text{Area} &= \left(\frac{276}{2}\right) \times \left(\frac{204}{2}\right) \\
\text{Area} &= 138 \times 102 \\
\text{Area} &= 14076 \text{ square meters}
\end{aligned}
$$

The representation of this mathematical practice can be explained by the following etic ethnomodel procedures: a) transform the shape of the irregular quadrilateral in a rectangle whose area can be determined through the application of the area formula, b) determine the dimensions of the rectangle by calculating the average of the two opposite sides of the irregular quadrilateral, and c) determine the area of the rectangle by applying the formula: $A = b \times h$.

<figure><img src="image_9.png" /></figure>
It is important, indeed relevant here to state that there is historical evidence that the method of cubação in which a quadrilateral is transformed into a rectangle was used with the purpose of land taxation in Ptolemaic and Roman periods, as well in ancient Egypt (Peet, 1970). This method is also used in Chile and Nepal, and in the Brazilian states of Bahia, Pernambuco, Rio Grande do Norte, Rio Grande do Sul, São Paulo e Sergipe (Silva, 2012).

Transforming the Shape of an Irregular Quadrilateral into a Square

The second approach is called Jorge's Method that is related to how to square the land, which means to transform the initial quadrilateral into a square with the same perimeter. Table 2 shows Adão's method of estimating an area of a land with irregular shape.

<table><caption>Table 2: Adão's method of estimating an area of a land with irregular shape. Source: Adapted from Knijnik (1993, p. 24)</caption><thead><tr><td>Adão's explanation (Emic knowledge)</td><td>Academic explanation (Etic knowledge)</td></tr></thead><tbody><tr><td>Here is a piece of land with four walls</td><td>This is a convex quadrilateral</td></tr><tr><td>First, we add all the walls</td><td>First, we determine the perimeter of this convex quadrilateral</td></tr><tr><td>Second, we divide the sum by four</td><td>Second, we divide the perimeter by four</td></tr><tr><td>Third, we multiply the obtained number by itself</td><td>Third, we determine the area of the square whose side is given by dividing its perimeter by four</td></tr><tr><td>This is the cubação of this land</td><td>This is the area of the square obtained from the perimeter of the convex quadrilateral</td></tr></tbody></table>

In this context, Jorge explained that “Since the land has four different sides [irregular shape], I added all four sides: 90, 124, 114, and 152 and the result is 480. Now, I divide this result by 4, which gives 120. Then, I multiply 120 by 120, which gives me 14400”. Thus, the quadrilateral is transformed into a square whose side is the fourth part of the perimeter of the original polygon.

<figure><img src="image_11.png" /></figure>

Thus, their emic mathematical knowledge can be represented by an etic ethnomodel that transforms the shape of the given land into a square of 120 metres each side.

$$ 
\begin{aligned}
\text{Area} &= \left( \frac{a + b + c + d}{4} \right) \\
\text{Area} &= \left( \frac{124 + 90 + 152 + 114}{4} \right)
\end{aligned}
$$
It is important, indeed relevant here to state that there is historical evidence that the method of cubação in which a quadrilateral is transformed into a rectangle was used with the purpose of land taxation in Ptolemaic and Roman periods, as well in ancient Egypt (Peet, 1970). This method is also used in Chile and Nepal, and in the Brazilian states of Bahia, Pernambuco, Rio Grande do Norte, Rio Grande do Sul, São Paulo e Sergipe (Silva, 2012).

Transforming the Shape of an Irregular Quadrilateral into a Square

The second approach is called Jorge's Method that is related to how to square the land, which means to transform the initial quadrilateral into a square with the same perimeter. Table 2 shows Adão's method of estimating an area of a land with irregular shape.

<table><caption>Table 2: Adão's method of estimating an area of a land with irregular shape. Source: Adapted from Knijnik (1993, p. 24)</caption><thead><tr><td>Adão's explanation (Emic knowledge)</td><td>Academic explanation (Etic knowledge)</td></tr></thead><tbody><tr><td>Here is a piece of land with four walls</td><td>This is a convex quadrilateral</td></tr><tr><td>First, we add all the walls</td><td>First, we determine the perimeter of this convex quadrilateral</td></tr><tr><td>Second, we divide the sum by four</td><td>Second, we divide the perimeter by four</td></tr><tr><td>Third, we multiply the obtained number by itself</td><td>Third, we determine the area of the square whose side is given by dividing its perimeter by four</td></tr><tr><td>This is the cubação of this land</td><td>This is the area of the square obtained from the perimeter of the convex quadrilateral</td></tr></tbody></table>

In this context, Jorge explained that “Since the land has four different sides [irregular shape], I added all four sides: 90, 124, 114, and 152 and the result is 480. Now, I divide this result by 4, which gives 120. Then, I multiply 120 by 120, which gives me 14400”. Thus, the quadrilateral is transformed into a square whose side is the fourth part of the perimeter of the original polygon.

<figure><img src="image_10.png" /></figure>

Thus, their emic mathematical knowledge can be represented by an etic ethnomodel that transforms the shape of the given land into a square of 120 metres each side.

$$ 
\begin{aligned}
\text{Area} &= \left( \frac{a + b + c + d}{4} \right) \\
\text{Area} &= \left( \frac{124 + 90 + 152 + 114}{4} \right)
\end{aligned}
$$
$$Area = \left(\frac{480}{4}\right)$$

$$Area = 120 \text{ square meters}$$

The representation of this mathematical practice can be explained by the following ethnomodel: a) transform the shape of the irregular quadrilateral in a rectangle whose area can be easily determined through the application of the area formula, b) determine the dimensions of the rectangle by calculating the average of the two opposite sides of the irregular quadrilateral, and c) determine the area of the square by applying the formula $A=a \times a = a^2$.

<figure><img src="image_12.png" /></figure>

In this context, Knijnik (1996) has affirmed that the methods used by Adão and Jorge are mathematical practices that rural workers in southern Brazil employ in order to transform irregular figures into regular ones. For example, in the *Adão's Method*, there is a reduction of the area of the land in a rectangular shape while in the *Jorge's Method* this area is reduced into a quadrangular shape. Nevertheless, it is important to state that the method applied by Jorge shows that there is an increase in area in relation to the method used by Adam because among all the quadrilaterals with the same perimeter, the square has the largest area.

## References

* D'Ambrosio, U. (1999) Literacy, matheracy, and technoracy: a trivium for today. *Mathematical Thinking and Learning, 1*(2), 131-153.

* Knijnik, G. (1993). An ethnomathematical approach in mathematical education: a matter of political power. *For the Learning of Mathematics, 13*(2), 23-25.

* Knijnik, G. (1996). *Exclusão e resistência: educação matemática e legitimidade cultural* [Exclusion and resistance: mathematics education and cultural legitimacy]. Porto Alegre, Rio Grande do Sul, Brazil: Artes Médicas, 1996.

* Lowrie, T. (2014). An educational practices framework: The potential for empowerment of the teaching profession. *Journal of Education for Teaching, 40*(1), 34-46.

* Peet, T. E. (1970). *The Rhind mathematical papyrus*. British Museum 10057 and 10058. Nendeln, Liechtenstein: Kraus Reprint.

* Rosa, M., Orey, D.C. (2016). State of the Art in Ethnomathematics. In: *Current and Future Perspectives of Ethnomathematics as a Program*. ICME-13 Topical Surveys. Springer, Cham. https://doi.org/10.1007/978-3-319-30120-4_3.

* Silva, G. R. (2012). Cultura e etnomatemática: apresentação de algumas estratégias matemáticas dos lavradores do assentamento rural Natur de Assis (BA) [Culture and ethnomathematics: presenting some mathematical strategies of peasants in the Natur de Assis (BA) settlement]. *Anais do 4o Congresso Brasileiro de Etnomatemática – CBEm4* (pp. 1-12). Belém, Pará, Brazil: UFBA/ABEM.
$$Area = \left(\frac{480}{4}\right)$$

$$Area = 120 \text{ square meters}$$

The representation of this mathematical practice can be explained by the following ethnomodel: a) transform the shape of the irregular quadrilateral in a rectangle whose area can be easily determined through the application of the area formula, b) determine the dimensions of the rectangle by calculating the average of the two opposite sides of the irregular quadrilateral, and c) determine the area of the square by applying the formula $A=a \times a = a^2$.

<figure><img src="image_11.png" /></figure>

In this context, Knijnik (1996) has affirmed that the methods used by Adão and Jorge are mathematical practices that rural workers in southern Brazil employ in order to transform irregular figures into regular ones. For example, in the *Adão's Method*, there is a reduction of the area of the land in a rectangular shape while in the *Jorge's Method* this area is reduced into a quadrangular shape. Nevertheless, it is important to state that the method applied by Jorge shows that there is an increase in area in relation to the method used by Adam because among all the quadrilaterals with the same perimeter, the square has the largest area.

## References

* D'Ambrosio, U. (1999) Literacy, matheracy, and technoracy: a trivium for today. *Mathematical Thinking and Learning, 1*(2), 131-153.

* Knijnik, G. (1993). An ethnomathematical approach in mathematical education: a matter of political power. *For the Learning of Mathematics, 13*(2), 23-25.

* Knijnik, G. (1996). *Exclusão e resistência: educação matemática e legitimidade cultural* [Exclusion and resistance: mathematics education and cultural legitimacy]. Porto Alegre, Rio Grande do Sul, Brazil: Artes Médicas, 1996.

* Lowrie, T. (2014). An educational practices framework: The potential for empowerment of the teaching profession. *Journal of Education for Teaching, 40*(1), 34-46.

* Peet, T. E. (1970). *The Rhind mathematical papyrus*. British Museum 10057 and 10058. Nendeln, Liechtenstein: Kraus Reprint.

* Rosa, M., Orey, D.C. (2016). State of the Art in Ethnomathematics. In: *Current and Future Perspectives of Ethnomathematics as a Program. ICME-13 Topical Surveys*. Springer, Cham. https://doi.org/10.1007/978-3-319-30120-4_3.

* Silva, G. R. (2012). Cultura e etnomatemática: apresentação de algumas estratégias matemáticas dos lavradores do assentamento rural Natur de Assis (BA) [Culture and ethnomathematics: presenting some mathematical strategies of peasants in the Natur de Assis (BA) settlement]. *Anais do 4o Congresso Brasileiro de Etnomatemática – CBEm4* (pp. 1-12). Belém, Pará, Brazil: UFBA/ABEM.
PLENARY PANEL 2
PLENARY PANEL 2
# TEACHING GEOMETRY IN FRANCE

Fabien Emprin¹ (chair of the panel), Isabelle Audra, Mélanie Binet, Marie-Paul Foy, Lucie Jacotin, Aurélie Marche.

¹ INSPé, University of Reims, Champagne-Ardenne

> It is traditional for ICMI Studies to enable participants to visit classes in the host country. This 26th ICMI Study took place in France during the school vacations, so we were able to organize a plenary session of discussion with French teachers. They will present their school context, their practices, and their questions to the group of researchers.

## AN OVERVIEW OF THE FRENCH EDUCATIONAL SYSTEM

### The main principles

The French educational system is guided by five principles, namely: academic freedom, free provision, neutrality, secularism, and compulsory education.

The freedom of education and relations with private education defines that parents can choose between public schools at no cost or private school subjected to state control, which can benefit from state aid [in return for a contract signed with the state] ("Debré Law" n°59-1557, December 31, 1959). Only the government is authorized to issue university diplomas and educational degrees. Teachers and students are expected to be philosophically and politically neutral. Neither religious instruction nor proselytizing are allowed. Education is compulsory for all children from 3 to 16 years old residing in France (the compulsoriness of school for the three-year-old child is recent, dating from September 2019). Regardless of their nationality, students must be educated, whether at school or at home (with a prior declaration).

### Student With Special Needs

French public schools aim at welcoming all children of all intellectual and physical abilities from the age of three in normal classes (disabled students; students with language and learning difficulties; students with long-term educational difficulties; gifted students). This schooling in mainstream education can be with or without specific support or arrangements according to the need of the student. A local service, the Departmental House for Disabled People (MDPH in French) can work out a personalized education plan for those children. Depending on the nature of a child's disability, they can either be homeschooled, taught in the local school within an educational inclusion unit or hosted in a specialized educational unit or at the hospital.

### Organization of the School Year

In metropolitan France, the school year runs from early September to early July. The school calendar is standardized but with variations within the regions (3 areas of vacation: for example, students of Créteil, or Toulouse are at work during ICMI Study, but Reims' students are not). Typically schools cycle around 7 weeks of work and 2 weeks of vacation. Summer vacation lasts approximately 2
# TEACHING GEOMETRY IN FRANCE

Fabien Emprin¹ (chair of the panel), Isabelle Audra, Mélanie Binet, Marie-Paul Foy, Lucie Jacotin, Aurélie Marche.

¹ INSPé, University of Reims, Champagne-Ardenne

It is traditional for ICMI Studies to enable participants to visit classes in the host country. This 26th ICMI Study took place in France during the school vacations, so we were able to organize a plenary session of discussion with French teachers. They will present their school context, their practices, and their questions to the group of researchers.

## AN OVERVIEW OF THE FRENCH EDUCATIONAL SYSTEM

### The main principles

The French educational system is guided by five principles, namely: academic freedom, free provision, neutrality, secularism, and compulsory education.

The freedom of education and relations with private education defines that parents can choose between public schools at no cost or private school subjected to state control, which can benefit from state aid [in return for a contract signed with the state] ("Debré Law" n°59-1557, December 31, 1959). Only the government is authorized to issue university diplomas and educational degrees. Teachers and students are expected to be philosophically and politically neutral. Neither religious instruction nor proselytizing are allowed. Education is compulsory for all children from 3 to 16 years old residing in France (the compulsoriness of school for the three-year-old child is recent, dating from September 2019). Regardless of their nationality, students must be educated, whether at school or at home (with a prior declaration).

### Student With Special Needs

French public schools aim at welcoming all children of all intellectual and physical abilities from the age of three in normal classes (disabled students; students with language and learning difficulties; students with long-term educational difficulties; gifted students). This schooling in mainstream education can be with or without specific support or arrangements according to the need of the student. A local service, the Departmental House for Disabled People (MDPH in French) can work out a personalized education plan for those children. Depending on the nature of a child's disability, they can either be homeschooled, taught in the local school within an educational inclusion unit or hosted in a specialized educational unit or at the hospital.

### Organization of the School Year

In metropolitan France, the school year runs from early September to early July. The school calendar is standardized but with variations within the regions (3 areas of vacation: for example, students of Créteil, or Toulouse are at work during ICMI Study, but Reims' students are not). Typically schools cycle around 7 weeks of work and 2 weeks of vacation. Summer vacation lasts approximately 2
months (July and August).

Most schools are open 4 days a week (Monday, Tuesday, Thursday, Friday). Some schools are open on Wednesday evening and secondary school often work on Wednesday and/or Saturday evening.

## Organization of scholarship

French school is divided into preschool, elementary school, middle school, high school and upper education including university, preparatory class for the “grandes écoles”, University Institute of Technology.

<table><caption>Table 1. French school system by student age</caption><thead><tr><th>Age</th><th>French name</th><th>French name/English name</th></tr></thead><tbody><tr><td>3</td><td>Petite section</td><td>Maternelle / Preschool</td></tr><tr><td>4</td><td>Moyenne section</td><td></td></tr><tr><td>5</td><td>Grande sections</td><td></td></tr><tr><td>6</td><td>Cours préparatoire — CP</td><td>Élémentaire / Primary school</td></tr><tr><td>7</td><td>Cours élémentaire première année — CE1</td><td rowspan="3">(École maternelle + élémentaire = école primaire)</td></tr><tr><td>8</td><td>Cours élémentaire seconde année — CE2</td></tr><tr><td>9</td><td>Cours moyen première année — CM1</td></tr><tr><td>10</td><td>Cours moyen seconde année — CM2</td><td></td></tr><tr><td>11</td><td>6e — sixième</td><td rowspan="4">Collège / Secondary school</td></tr><tr><td>12</td><td>5e — cinquième</td></tr><tr><td>13</td><td>4e — quatrième</td></tr><tr><td>14</td><td>3e — troisième</td></tr><tr><td>15</td><td>2de — seconde</td><td>Lycée / High school</td></tr><tr><td>16</td><td>1ère — première</td><td rowspan="2">(Collège + lycée = enseignement secondaire)</td></tr><tr><td>17</td><td>Terminale</td></tr><tr><td></td><td>Baccalauréat</td><td>Examen / exam</td></tr><tr><td></td><td>Licence (3 years)</td><td>University grades</td></tr><tr><td></td><td>Master (2 years)</td><td rowspan="2">University post-grades</td></tr><tr><td></td><td>Doctorat (3 years)</td></tr></tbody></table>

## Teachers' education and recruitment

In France, teachers are public servants. They must pass a competitive entrance examination. There are three possible examinations: CAPE (certificate of aptitude to teach in preschool or primary school), CAPES (certificate of aptitude to teach in secondary school) and Agrégation which also allows teaching in secondary school (but also to some extent in preparatory classes and university). All teachers must have a master's degree.

Students with a bachelor's degree can enroll in a master's degree in teaching, education, and training (MEEF) at an Institut Supérieur du Professorat et de l'Éducation (INSPÉ). At the INSPÉ they prepare
months (July and August).

Most schools are open 4 days a week (Monday, Tuesday, Thursday, Friday). Some schools are open on Wednesday evening and secondary school often work on Wednesday and/or Saturday evening.

## Organization of scholarship

French school is divided into preschool, elementary school, middle school, high school and upper education including university, preparatory class for the "grandes écoles", University Institute of Technology.

<table><caption>Table 1. French school system by student age</caption><thead><tr><th>Age</th><th>French name</th><th>French name/English name</th></tr></thead><tbody><tr><td>3</td><td>Petite section</td><td>Maternelle / Preschool</td></tr><tr><td>4</td><td>Moyenne section</td><td></td></tr><tr><td>5</td><td>Grande sections</td><td></td></tr><tr><td>6</td><td>Cours préparatoire — CP</td><td>Élémentaire / Primary school</td></tr><tr><td>7</td><td>Cours élémentaire première année — CE1</td><td rowspan="3">(École maternelle + élémentaire = école primaire)</td></tr><tr><td>8</td><td>Cours élémentaire seconde année — CE2</td></tr><tr><td>9</td><td>Cours moyen première année — CM1</td></tr><tr><td>10</td><td>Cours moyen seconde année — CM2</td><td></td></tr><tr><td>11</td><td>6e — sixième</td><td rowspan="4">Collège / Secondary school</td></tr><tr><td>12</td><td>5e — cinquième</td></tr><tr><td>13</td><td>4e — quatrième</td></tr><tr><td>14</td><td>3e — troisième</td></tr><tr><td>15</td><td>2de — seconde</td><td>Lycée / High school</td></tr><tr><td>16</td><td>1ère — première</td><td rowspan="2">(Collège + lycée = enseignement secondaire)</td></tr><tr><td>17</td><td>Terminale</td></tr><tr><td></td><td>Baccalauréat</td><td>Examen / exam</td></tr><tr><td></td><td>Licence (3 years)</td><td>University grades</td></tr><tr><td></td><td>Master (2 years)</td><td rowspan="2">University post-grades</td></tr><tr><td></td><td>Doctorat (3 years)</td></tr></tbody></table>

## Teachers' education and recruitment

In France, teachers are public servants. They must pass a competitive entrance examination. There are three possible examinations: CAPE (certificate of aptitude to teach in preschool or primary school), CAPES (certificate of aptitude to teach in secondary school) and Agrégation which also allows teaching in secondary school (but also to some extent in preparatory classes and university). All teachers must have a master's degree.

Students with a bachelor's degree can enroll in a master's degree in teaching, education, and training (MEEF) at an Institut Supérieur du Professorat et de l'Éducation (INSPÉ). At the INSPÉ they prepare
for a master's degree and for the competitive examination.

## Some Peculiarities of the French System

Schools do not recruit teachers. The system for assigning teachers to their posts is governed by a national and regional system based on the allocation of points according to seniority, career, and family situation.

The *agrégation* allows teachers to teach at the same level as the CAPES, but an “*agrégé*” teacher teaches fewer hours per week (15h vs. 18h) and at a higher salary.

In higher education, there are several stages in the process of obtaining a research position. First, you need to obtain a PhD, and then apply to a section of the national university committee (CNU). This involves putting together a dossier outlining how your research fits into a particular disciplinary field. There are two CNU sections for math education researchers: Applied Mathematics and Application of Mathematics (26th section) and Education and Training Sciences (70th CNU section). Only PhDs who have obtained this national qualification are eligible to take part in competitive examinations for positions as lecturers (associate professor).

To become a full professor, you need to obtain a second diploma, the “*habilitation à diriger des recherches*” (habilitation to supervise research), and then apply for a position. Qualification has existed in the same way as for lecturers but is no longer compulsory for associate professors working at a French university.

## THE FRENCH SYSTEM IN FIGURES

### Average salary (in July 2021, source Ministry of Education):

* Certified teachers: 2490 euros net per month. Time in front of pupils: 18 h per week

* Primary school teachers: 2407 euros net per month. Time in front of pupils: 24h per week

* Agrégé: 3719 euros net per month. Time in front of students: 15 h per week

* Lecturer: 2805 euros net per month. Time in front of students: 192 h/year

* University professor: 4200 euros net per month. Time in front of students: 192 h/year

### Pupils:

* 11,997,900 schoolchildren (public and private, forecast for the start of the 2023 school year)

* 6,349,600 primary school pupils

* 5,648,300 secondary school students, including 3,397,300 middle school students and 2,251,000 high school students (including 627,100 vocational high school students)

### Schools:

* 58,910 public and private secondary schools under contract (back to 2022) [48,220 schools; 6,980 collèges]

* 3,710 lycées and Erea (*rétablissement régional d'enseignement adapté*), including 2,080 lycées for vocational high school students

### Teachers:

* 853,700 national education employees teach 1st and 2nd grade students (public and private under contract, back to 2022)
for a master's degree and for the competitive examination.

## Some Peculiarities of the French System

Schools do not recruit teachers. The system for assigning teachers to their posts is governed by a national and regional system based on the allocation of points according to seniority, career, and family situation.

The *agrégation* allows teachers to teach at the same level as the CAPES, but an “*agrégé*” teacher teaches fewer hours per week (15h vs. 18h) and at a higher salary.

In higher education, there are several stages in the process of obtaining a research position. First, you need to obtain a PhD, and then apply to a section of the national university committee (CNU). This involves putting together a dossier outlining how your research fits into a particular disciplinary field. There are two CNU sections for math education researchers: Applied Mathematics and Application of Mathematics (26th section) and Education and Training Sciences (70th CNU section). Only PhDs who have obtained this national qualification are eligible to take part in competitive examinations for positions as lecturers (associate professor).

To become a full professor, you need to obtain a second diploma, the “*habilitation à diriger des recherches*” (habilitation to supervise research), and then apply for a position. Qualification has existed in the same way as for lecturers but is no longer compulsory for associate professors working at a French university.

## THE FRENCH SYSTEM IN FIGURES

### Average salary (in July 2021, source Ministry of Education):

* Certified teachers: 2490 euros net per month. Time in front of pupils: 18 h per week

* Primary school teachers: 2407 euros net per month. Time in front of pupils: 24h per week

* Agrégé: 3719 euros net per month. Time in front of students: 15 h per week

* Lecturer: 2805 euros net per month. Time in front of students: 192 h/year

* University professor: 4200 euros net per month. Time in front of students: 192 h/year

### Pupils:

* 11,997,900 schoolchildren (public and private, forecast for the start of the 2023 school year)

* 6,349,600 primary school pupils

* 5,648,300 secondary school students, including 3,397,300 middle school students and 2,251,000 high school students (including 627,100 vocational high school students)

### Schools:

* 58,910 public and private secondary schools under contract (back to 2022) [48,220 schools; 6,980 collèges]

* 3,710 lycées and Erea (rétablissement régional d'enseignement adapté), including 2,080 lycées for vocational high school students

### Teachers:

* 853,700 national education employees teach 1st and 2nd grade students (public and private under contract, back to 2022)