congress of the European society for research in mathematics education (pp. 21-30). ERME and Universitat Ramon Llull.

CHEVALLARD, Y., (2015). Teaching Mathematics in Tomorrow's Society: A Case for an Oncoming Counter Paradigm. In S. Cho (Ed.), *The Proceedings of the 12th International Congress on Mathematical Education* (pp. 173-188). Springer. https://doi.org/10.1007/978-3-319-12688-3_13

CHEVALLARD, Y. (2019). Introducing the Anthropological Theory of the Didactic: an attempt at a principled approach. *Hiroshima Journal of Mathematics Education, 12*, 71-114. https://doi.org/10.24529/hjme.1205

CHEVALLARD, Y. (2021). La question curriculaire à la lumière de la TAD : défigement praxéologique et questionnement du monde [The curricular question in the light of the ATD: praxeological disfigurement and questioning the world]. In H. Chaachoua et al. (Eds.), *Actes XXe école d'été de didactique des mathématiques* (pp. 93-112). La pensée sauvage.

CHEVALLARD, Y., & BOSCH, M. (2014). Didactic Transposition in Mathematics Education. In S. Lerman (Eds.), *Encyclopedia of Mathematics Education* (pp. 170-174). Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4978-8_48

DINDYAL, J. (2020). International comparative studies in mathematics: An overview. In S. Lerman (Ed.), *Encyclopedia of mathematics education* (pp. 320–325). Springer. https://doi.org/10.1007/978-3-030-15789-0_83

HONG, D.S., & CHOI, K.M. (2014). A comparison of Korean and American secondary school textbooks: the case of quadratic equations. *Educational Studies in Mathematics*, 85, 241–263. https://doi.org/10.1007/s10649-013-9512-4

JABLONKA, E., ANDREWS, P., CLARKE, D., & XENOFONTOS, C. (2018). Comparative studies in mathematics education: Twenty years of communication, cooperation and collaboration in Europe. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), *Developing research in mathematics education: Twenty years of communication, cooperation and collaboration in Europe* (pp. 223–238). Routledge.

KASPARY, D., CHAACHOUA, H., & BESSOT, A. (2020). Qu’apporte la notion de portée d’une technique à l’étude de la dynamique praxéologique? [What does the notion of technique reach bring to the study of praxeological dynamics?] *Annales de didactique et de sciences cognitives*, 25, 243-269. https://doi.org/10.4000/adsc.573

MULLIS, I.V.S., & MARTIN, M.O. (Eds.). (2015). *TIMSS 2015 assessment frameworks*. Retrieved from Boston College, TIMSS & PIRLS International Study Center website: https://timssandpirls.bc.edu/timss2015 framewor k.html

RAVEL, L. (2002). *Des programmes à la classe : étude de la transposition didactique interne. Exemple de l’arithmétique en Terminale S spécialité mathématique* [Doctoral dissertation, Université Joseph Fourier]. https://theses.hal.science/tel-00162790
**Appendix 2 – Chronogenesis of the techniques in each country**

<figure><img src="image_11.png" /><figcaption>France, Grade 11</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Spain, Grade 8</figcaption></figure>

<figure><img src="image_13.png" /></figure>
SON, J.-W., WATANABE, T., & LO, J.-J. (2017). *What matters? Research Trends in International Comparative Studies in Mathematics Education*. Research in Mathematics Education. Springer Nature.

TRAVERS, K. (1992). Overview of the longitudinal version of the second international study. In L. Burstein (Ed.), The IEA study of mathematics III: Student growth and classroom processes (pp. 1–4). Pergamon Press. https://doi.org/10.1016/B978-0-08-041371-6.50009-2

WIJAYANTI, D., & BOSCH, M. (2018). The evolution of the knowledge to be taught through educational reforms: The case of proportionality. In Y. Shimizu & R. Vithal (Eds.), *School mathematics curriculum reforms: Challenges, changes and opportunities. Proceedings of the twenty-fourth ICMI study conference* (pp. 173–180). ICMI.

WU, M. (2006). A comparison of mathematics performance between East and West: What PISA and TIMSS can tell us. In F. K. S. Leung, K.-D. Graf, & F. J. Lopez-Real (Eds.), *Mathematics education in different cultural traditions: A comparative study of East Asia and the West* (pp. 227–238). Springer. https://doi.org/10.1007/0-387-29723-5_15

ZHU, Y., & FAN, L. (2006). Focus on the Representation of Problem Types in Intended Curriculum: A Comparison of Selected Mathematics Textbooks from Mainland China and the United States. *Int J Sci Math Educ, 4*, 609–626. https://doi.org/10.1007/s10763-006-9036-9

## Brazilian Textbooks

GIOVANNI JÚNIOR, J. R., & CASTRUCCI, B. (2018). *A conquista da Matemática. Grade 8*. FTD.

GIOVANNI JÚNIOR, J. R., & CASTRUCCI, B. (2018). *A conquista da Matemática. Grade 9*. FTD.

SOUZA, J. R. (2018). *Matemática realidade & tecnologia. Grade 8*. FTD.

SOUZA, J. R. (2018). *Matemática realidade & tecnologia. Grade 9*. FTD.

BIANCHINI, E. (2018). *Matemática – Bianchini. Grade 8*. Moderna.

BIANCHINI, E. (2018). *Matemática – Bianchini. Grade 9*. Moderna.

## French textbooks

BARNET, C., BERGER, H., BILLA, N., DEMOULIN, P., FLOUS, A., LAFARGUE, B., LARRIEU, M., LAULHERE, A., LAYAN M.-C., POLLET, S., ROBERTOU, M., RUDELLE, F., VILLATTES, A. (2016). *Indigo, Maths, Cycle 4, Troisième*. Hachette.

BOULLIS, M., CAMBON, M., DANARD, Y., GALLIEN, V., HERRMANN, É., MONKA, Y., & PERCOT, S. (2016). *Myriade, Maths, Cycle 4, Troisième*. Bordas.

LE YAOUANQ, M.-H., BEAUVOIT, E., GASTIN, H., GUILLET, D., SALIBA, G., SOMMIER, F., & TADEUSZ, L. (2015). *Math'x, Première S*. Didier.

BERHOUET, J., BRO, F., CORNE, R., DELPUECH, C., LE YAOUANQ, M.-H., & PAULOU, F. (2019). *Math'x, Seconde*. Didier.

CARLOD, V. (2016). *Transmath, Troisième*. Paris : Nathan.

LIXI, C. (2019). *Transmath, Seconde*. Nathan.

DHOMBRES, J., PAINTOUX, T., LOUBATIERES, J., PIAU, Y., & CROC, C. (2019). *Transmath, Première*. Nathan.
<figure><img src="image_14.png" /><figcaption>Vietnam, Grade 9</figcaption></figure>

<figure><img src="image_15.png" /></figure>
PONCY, M., & VIEUDRIN, D. (2019). *Indice, Maths, Première, enseignement de spécialité*. Bordas.

PONCY, M., & VIEUDRIN, D. (2019). *Indice, Maths, Seconde*. Paris : Bordas.

## Japanese Textbooks

OKABE, T. (2015). *Chugakko Sugaku 3* [中学校数学3]. Suken Shuppan.

OKAMOTO, H. (2015a). *Mirai e Hirogaru Sugaku 3* [未来へひろがる数学3]. Keirinkan.

OKAMOTO, H. (2015b). *Gateway to the future Math 3: For Junior High School*. Keirinkan. (English edition)

FUJII, T (2015). *Atarashii Sugaku 3* [新しい数学3]. Tokyo Shoseki.

## Spanish Texbooks

AA.VV (2016). *Matemátiques 2 ESO*. Edebé.

AA.VV (2015). *Matemátiques 3 ESO*. Edebé.

AA.VV (2016). *Matemátiques 4 ESO*. Edebé.

AA.VV (2015). *Matemàtiques, Sèrie RESOL, 2º Secundària, Català ED 2015*. Santillana.

AA.VV (2015). *Matemàtiques, Sèrie RESOL, 3º Secundària, Català ED 2015*. Santillana.

AA.VV (2016). *Matemàtiques, Sèrie RESOL, 4º Secundària, Català ED 2015*. Santillana.

## Vietnamese Textbook

PHAM BAO KUÉ & AL (2011). *Grade 9, Vol. 2*. Publisher: Ministry of Education. (Translated from Vietnamese)
<table><caption>**APPENDIX 1 – A priori** description of the sub-types of tasks $T_i$ and their pragmatic scope</caption><thead><tr><th>Sub-type of tasks</th><th>Generic expression of T<sub>i</sub></th><th>Pragmatic scope</th></tr></thead><tbody><tr><td>T<sub>1</sub></td><td>$ax^2 = c, ac > 0$</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>2</sub></td><td>$[nk^2]x^2 = [nk'^2]$</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>3</sub></td><td>$[nk^2]x^2 - [nk'^2] = 0$</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>4</sub></td><td>$ax^2 + c = 0; ac < 0$ [C]</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>5</sub></td><td>$ax^2 + c = 0; ac > 0$ [C]</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>6</sub></td><td>$ax^2 + bx = 0$ [C-P]</td><td>$\tau_{null_product}$</td></tr><tr><td>T<sub>7</sub></td><td>$ax^2 + bx + c = 0$ [C-P]</td><td>$\tau_{discriminant}$</td></tr><tr><td>T<sub>8</sub></td><td>$ax^2 + bx + c = 0; a \pm b \pm c = 0$ or $4a \pm 2b \pm c = 0$ or $9a \pm 3b \pm c = 0$</td><td>$\tau_{discriminant}$</td></tr><tr><td>T<sub>9</sub></td><td>$[ka^2]x^2 \pm [2kac]x + [kc^2] = 0$ [C-P]</td><td>$\tau_{null_product}$</td></tr><tr><td>T<sub>10</sub></td><td>$k(ax + b)^2 = k'; kk' > 0$ [C-P]</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>11</sub></td><td>$k(ax + b)^2 = k'; kk' < 0$ [C-P]</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>12</sub></td><td>$[nk^2](ax + b)^2 - [nk'^2] = 0$ [C]</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>13</sub></td><td>$[nk^2](ax + b)^2 = [nk'^2]$</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>14</sub></td><td>$[nk^2](ax + b)^2 + [nk'^2] = 0$ [C]</td><td>$\tau_{square_root}$</td></tr><tr><td>T<sub>15</sub></td><td>$[nk^2](ax + b)^2 - [nk'^2](cx + d)^2 = 0$ [C-P]</td><td>$\tau_{null_product}$</td></tr><tr><td>T<sub>16</sub></td><td>$k(ax + b)(cx + d)(...)= 0$</td><td>$\tau_{null_product}$</td></tr><tr><td>T<sub>17</sub></td><td>$k(ax + b)(cx + d) + k'(nax + nb)(ex + f) + \cdots = 0$ [C-P]</td><td>$\tau_{null_product}$</td></tr><tr><td>T<sub>18</sub></td><td>$k(ax + b)(cx + d) + [k'a^2]x^2 - [k'b^2] = 0$ [C-P]</td><td>$\tau_{null_product}$</td></tr><tr><td>T<sub>19</sub></td><td>$P_2(x) = Q(x)$, with apparent factor after partial factorization [P]</td><td>$\tau_{null_product}$</td></tr><tr><td>T<sub>20</sub></td><td>$P_2(x) = Q(x)$, without apparent factorization [P]</td><td>$\tau_{discriminant}$</td></tr></tbody></table>

The term [C] means “commutation” of the terms of the equation and [P] “permutation” of the terms with respect to the “=”. For example, for T4, the [C] indicates that $ax^2 + c = 0$ et $c + ax^2 = 0$ belong to the same generic expression $ax^2 + c = 0$ with $ac < 0$.
**APPENDIX 2 – *Chronogenesis* of the techniques in each country**

<caption>Brazil, Grade 9</caption>

Square Root
Nul Product
Square Root
Nul Product
Square Root
Nul Product
Square Root
Discriminant
Nul Product
SP_by trial
Discriminant

<figure><img src="image_5.png" /><figcaption>France, Grade 11</figcaption></figure>

Nul Product
Square Root
SP_Evident Root
Nul Product
SP_Evident Root
SP_Evident Root
SP_Evident Root
SP_Evident Root
Discriminant

<figure><img src="image_6.png" /></figure>
<figure><img src="image_7.png" /><figcaption>Japan, Grade 9</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Vietnam, Grade 9</figcaption></figure>

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