Plenary and Special Lectures Program
In accordance to our conference theme "Entre Culturas / Across Cultures", we have invited plenary and special speakers that represent different cultural contexts, as well as forms of research in Mathematics Education. They come from different countries, not only in NorthAmerica, but also from Europe. They represent different areas of research and disciplines within Mathematics Education; and we include researchers from different generations. We hope that these lectures will reflect parts of the ample diversity of ways of teaching and learning of mathematics, and of the tools and communities involved in mathematics education. Through all of the above, we hope to honor the major goals of PMENA
Plenary Lectures
Plenary Panel: Reflections on Digital Technologies in Mathematics Education Across Cultures
Celia Hoyles

Dame Celia Hoyles is Professor of Mathematics Education at the UCL Institute of Education, Knowledge Lab, London, England.
 First recipient of the International Commission of Mathematics Instruction (ICMI) Hans Freudenthal medal in 2004 for excellence in research.
 First recipient of the Royal Society Kavli Education Medal in 2011 for excellence in research.
 Hon Doctorate of Philosophy, the Open University; Hon Doctorate of Science, Loughborough University; Hon Doctorate of Sheffield Hallam University (Hon DUniv)
 She was the UK Government's Chief Adviser for mathematics (2004 07), and the director of the National Centre for Excellence in the Teaching of Mathematics (200713). She was President of the Institute of Mathematics and its Applications (IMA) (201415).
 Celia was awarded an OBE in 2004 and made a Dame Commander of the Order of the British Empire in 2014.
Her major research interests are: secondary students' conceptions of proof, the mathematical skills needed in modern workplaces and the design of computer environments to engage students in learning mathematics. 
Carolyn Kieran

Carolyn Kieran is Professor Emerita of the Université du Québec à Montréal, Canada. Her research has been devoted to the study of algebra teaching and learning. Her interest lies in emphasizing both the conceptual and the technical dimensions of algebra, in encouraging the design of tasks that allow students to become aware of the conceptual aspects of algebra while engaging in the technical, and in investigating the ways in which the technical and the conceptual can coemerge in algebra learning. Computing technology, in particular that of Computer Algebra Systems (CAS), have been a central component of some of her research. Prior to her retirement, she was head of the research group, Algebra in Partnership with Technology in Education (APTE), and has been principal investigator, or coinvestigator, of 27 funded projects, published in over 200 articles and book chapters. Her contributions to the international mathematics education research community include serving as Chair of the ICMI Awards Committee (20112016), as President of the International Group for the Psychology of Mathematics Education (PME) and as member of the Board of Directors of the National Council of Teachers of Mathematics (NCTM).

Teresa Rojano

Teresa Rojano is Professor Emerita of the Department of Mathematics Education of the Center for Advanced Studies and Research (Cinvestav) in Mexico City, Mexico. She has conducted research in algebraic thought with a special focus on the transition from arithmetic to algebra. Her most recent projects focus on a web environment with artificial intelligence and adaptative system to investigate feedback models in parameterized modeling activities and the development of the meaning of the structure in algebra. She was vicepresident of the International Group for the Psychology of Mathematics Education –PME (19951997); member of the Program Committee of the International Congress of Mathematics Education – ICMI (20012004); leader (with Luis Puig) of the History Group for the Algebra ICMI Study; and director of the Mexican project Incorporation of New Technologies to the School Culture. She was head of the Department of Mathematics Education in Cinvestav (19972003); advisor of the Mexican Ministry of Education (SEP) (20032006); and academic advisor for the “New Model for the Telesecondary School” project (Latin American Institute of Educational Communication ILCESEP). Member of the Advisory Board of the James J. Kaput Center for Research and Innovation in Mathematics Education (UMASSD) since 2008. And member of several advisory and editorial boards (e.g., of the Journal for Research in Mathematics Education (NCTM), Educational Studies in Mathematics, and Mathematical Thinking and Learning).

María Trigueros
(Moderator)

María Trigueros has worked for 37 years at the Mathematics Department in Instituto Tecnológico Autónomo de México (ITAM), México, Maria studied Physics at the Universidad Nacional Autónoma de México (UNAM) and a PhD in Education at Universidad Complutense de Madrid, Spain. She is a member of the National Academy of Sciences and of the National Researchers System in México. Her research interests include the use of technology in the teaching of Mathematics, the learning and teaching of variable in elementary algebra and the learning and teaching of mathematics at university level. In this last area she is a leading researcher on the use of APOS theory.
María has participated in important Mexican national projects on the use of technology in the teaching of mathematics such as EFIT and EMAT (Teaching Physics with Technology and Teaching Mathematics with Technology) projects. She also coordinated the Mathematics group of the National Project Enciclomedia where she contributed to the development of programs to teach mathematics to elementary school children.
María was twice an elected member of the PMENA steering committee (20002003 and 20032006).

We are asking each of the three panelists to give us their reflections and perspectives on the following questions related to technology integration in Mathematics Education across cultures:
 In which ways has the digital technology culture evolved in our respective school systems (e.g., curricula, professional development, types of tools, student participation, etc) since the 1980s when the Logo movement began?
 What are your reflections on the potentials (and whether these have been harnessed) and challenges/obstacles, pros and cons, that you have seen from using digital technologies:
 For enhancing the learning of mathematics – in particular subject domains?
 For engaging more students/people of all ages and genders?
 For rethinking the mathematics curriculum at different levels (i.e., from 6 16; highschool, university)
 For assessment/evaluation?
 Of mathsspecific software and/or programming? And of the rapid rise of coding/programming?
 Based on your own research and possibly that of others within your respective countries, can you give some examples, which serve to illustrate how the use of digital technologies can foster enhanced engagement in mathematical thinking, particularly among groups who traditionally have been excluded (and as appropriate, highlighting cultural differences across our three countries, drawn from these examples)?
 In which ways outcomes from studies in mathematics education that involved the use of technology learning environments have influenced the mathematics curriculum design and development, as well as mathematical practices in and out of school across different cultures/countries?
 Do the features of ‘computational thinking’, which began with Papert and has become a movement characterized in different ways by Wing (2006) and others, relate to the processes of mathematical practice and, if so, in which ways? What are the relationships and differences between mathematical thinking and computational thinking?
 In which ways do research experiences serve as windows to observe or identify features of the use of technology devices and technology learning environments in different mathematics school cultures?
 What do you consider are the needs of professional learning for teachers to effectively teach with technology?
Plenary Session: Neuro and Cognitive Sciences in Mathematics Education
Educational neuroscience: Past, present and future prospects
Stephen Campbell

Stephen R. Campbell. Campbell is Associate Professor in the Faculty of Education at Simon Fraser University in Greater Vancouver, British Columbia, Canada. His areas of interest include mathematics education, the philosophy of education, and educational neuroscience. More specifically, Dr. Campbell’s scholarly focus is on the historical and psychological development of mathematical thinking from an embodied perspective informed by Kant, Husserl, and MerleauPonty. His research incorporates methods of psychophysics and cognitive neuroscience as a means for operationalising affective and cognitive models of math anxiety and concept formation. 
This talk will provide the speaker’s perspective on how the fledgling new area of educational neuroscience has emerged from a disenchantment with brainbased education, through various multidisciplinary, interdisciplinary, and transdisciplinary initiates and collaborations involving educationists and neuroscientists. Specific examples and results pertaining to research in mathematics education will be presented. Beyond the current stateoftheart, the speaker will conclude with some speculations on what might be anticipated as this area of research continues to unfold into the near and far futures.
Eyetracking mathematical representations – finding links between vision and reasoning
Ulises Xolocotzin

Ulises Xolocotzin is a Full Researcher at the Department of Mathematics Education of the Center for Advanced Studies and Research (Cinvestav) in Mexico City, Mexico. With a 2010 PhD in Psychology from the University of Nottingham, UK and a postdoc (20102012) at University of Bristol, UK, Ulises does research in Educational Technology, Mathematics Education and Educational Psychology. He has focused in early algebra and cognitive development; emotions in mathematics education a theme in which he edited an Elsiever book in 2017: Understanding Emotions in Mathematical Thinking and Learning; and eyetracking technologies to find links between between vision and mathematical reasoning.

Eyetracking studies need adequate theoretical frameworks for producing insights about mathematics learning. The current study uses eyetracking to investigate the effectiveness of tables and diagrams for supporting covariational reasoning amongst elementary students (n = 60). The theoretical framework emphasizes the cognitive functions of representations. Students showed more covariational reasoning around diagrams. The fixations showed that tables concentrated students’ attention on the dependent variable data, whereas diagrams distributed students’ attention evenly across the numeric and visual elements of the task. According to the theoretical framework, tables did not constrain a covariational interpretation of numerical data, whereas diagrams effectively constrained covariational interpretations, disrupting recursive tendencies and promoting the construction of a mental model of covariation.
Special Lectures
Indigenous people and mathematics education in Mexico
Pueblos indígenas y educación matemática en México
Natalia de Bengoechea

Natalia de Bengoechea studied a bachelor's and a master's degree in mathematics at the Faculty of Sciences of the National Autonomous University of Mexico, a Diploma in Descriptive Linguistics and Cultural Empathy at the Madero University of Puebla, and a Master's Degree in Mathematics Didactics at the University of Granada, Spain. She is interested in mathematics education, in particular ethnomathematics, and has published various articles and books and has led work groups and projects in mathematics education. She has been a professor at the National Pedagogical University of Mexico since 1979.

In Mexico, cultural diversity is present mainly through its indigenous peoples. Manifestations of mathematics, prior to the colonial era, of our native peoples are known. Many of that knowledge and manifestations have been lost or have remained in the vestiges of those civilizations. However, the inheriting peoples of these cultures continue to create and recreate implicit mathematics in their works and also retain some of the ancient knowledge in use. Languages and everyday activities are the main carriers of them. Education plays an important role in the preservation and retransmission of knowledge. Research is required that allows education to work on the knowledge of these peoples. Examples of analysis of indigenous numbers carried out for this purpose are presented.
En México la diversidad cultural está presente principalmente a través de sus pueblos indígenas. Se conocen manifestaciones de las matemáticas de nuestros pueblos originarios antes de la época colonial. Muchos de esos conocimientos y manifestaciones se han perdido o han quedado en los vestigios de esas civilizaciones. Sin embargo, los pueblos herederos de esas culturas siguen creando y recreando matemáticas implícitas en sus obras y también conservan en uso algunos de los conocimientos antiguos. Las lenguas y las actividades cotidianas son las principales portadoras de ellos. La educación juega un papel importante en la conservación y retransmisión de los conocimientos. Se requiere investigación que permita a la educación trabajar los conocimientos de estos pueblos. Se presentan ejemplos de análisis de numeraciones indígenas realizados con este fin.
Understanding teachers’ professional development through their interaction with resources: a multilingual project
Luc Trouche

Luc Trouche is Professor Emeritus of the Institut Français de l'Éducation  École Normale Supérieure de Lyon, France.
 Former President of the French Commission on Mathematics Teaching (CFEM)
 Former Head of the Department of Research of the IFÉENSL
 Former Chair of the Department Education and Technology of the French National Institute of Pedagogical Research.
 Fomer director of the IREM  Research Institute on Mathematics Teaching, Université Montpellier 2
Research interests: His work was previously dedicated to the didactical study of ICT integration in Mathematics Education, considering the interplay between instrumentation and conceptualization processes. This work inspired him to study, particularly, the teacher's role, introducing the notion of orchestration to model the didactical management of available artefacts in a classroom. He then focused on the design of resources and the teacher training required in the process of digitalization. This led him, in a joint work with Ghislaine Gueudet, to introduce the Documentational Approach To Didactics, considering all the resources involved in the teacher's activity, and analyzing teacher professional development as an interplay between practice, individual and collective, and resources; and where teacher collective work plays an essential role, both for renewing teacher resources and for fostering teacher professional development. 
The Documentational Approach To Didactics (DAD) aims to study teachers’ professional development through their interactions with their resources for/from teaching. It has been introduced in the French community of didactics of mathematics in 2007, then extended at an international level. It has been introduced as an entry in the Encyclopedia of Mathematics Education in 2020. The DADMultilingual project (20202021), is dedicated to gather and confront the translations of this entry towards several languages (Arabic, Chinese, French, German, Greek, Hebrew, Hungarian, Italian, Japanese, Norwegian, Portuguese, Spanish, Turkish and Ukrainian). The project main goals are: • Making available a presentation of DAD in various languages, allowing the researchers interested in using this approach to refer to in their own language; • Deepening the DAD concepts themselves in thinking their possible instantiations in different languages, and questioning the translating process itself; • Beyond the frame of DAD, questioning the notion of resource itself, resource for/from teaching. The lecture will present this project, and draw some lessons from its first steps.
Mental mathematics in the classroom: Content, practices and Papert’s Mathland
Photo (c) N.StPierre UQAM
Jérôme Proulx

Jérôme Proulx is Professor of Mathematics Education at the Université du Québec à Montréal, Canada. He runs the Laboratoire Épistémologie et Activité Mathématique. His research focuses on epistemological issues related to cognition and/in mathematics. His current research programme is dedicated to mental mathematics and school mathematics content development. 
This lecture reports on aspects of a larger research programme focused on studying mental mathematics in elementary and secondary mathematics classrooms. It specifically addresses an unplanned aspect that became salient through the work conducted in these classrooms. In this research programme, mental mathematics sessions are designed on a variety of mathematical topics (e.g., algebra, geometry, measurement, statistics, trigonometry, fractions), where students are given short amounts of time to solve tasks given orally and/or on the board, without the use of paperandpencil or any material aids. Whereas the central objectives centers on inquiring into the nature of the strategies students engage in to solve the tasks, more seem to be happening in these sessions. In particular, students’ solutions and strategies to the task given in the mental mathematics context led to numerous questions, discussions, followup explorations, and so forth, by students, which in turn enabled the emergence of significant mathematical issues. This raised interest in investigating these (additional and unplanned) mathematical issues. This represents the core of this lecture, which focuses on the nature of the mathematics (in terms of content and of practices) that frequently unfolds in the mental mathematics sessions conducted. Using an illustrative extract from a mental mathematics session on analytical geometry in a Grade10 classroom (1516 years old), the analysis outlines how not only mathematical content is being worked on through these mental mathematics sessions, but also how mathematical practices are being enacted by students. This raises issues about the nature of the environment that these mental mathematics session plunge students into, one that could be tentatively, and boldly, aligned with Papert’s concept of mathland.
Construction of ArithmeticAlgebraic Thinking in a SocioCultural Instructional Approach
Fernando Hitt

Fernando Hitt is Professor in the Department of Mathematics of the Université du Québec à Montréal, Canada. His research interests include the analysis of epistemological obstacles, the role of representations in the learning and teaching of mathematics and the use of technology in mathematics education. He was the cochair of the 1999 PMENA 21 conference in Cuernavaca, Mexico and leader of the PMENA Working Group (1998‐2002) Representations and Mathematics Visualization which resulted in a book of the same name. 
This talk will present the results of a research project on arithmeticalgebraic thinking that was carried out jointly by a team in Mexico and another in Quebec. The project deals with the concepts of variable and covariation between variables in the sixth grade at the elementary level and the first, second, and third years of secondary school – namely, children from 11 to 14 years old. We target secondary students (first year or K7) in this article. Our objective relates to the development of a gradual generalization in arithmeticalgebraic thinking in a sociocultural approach to the learning of mathematics. We experimented with investigative situations using a paperandpencil approach and technology. We analyze the emergence, in this context, of a visual abstraction, the production of institutional and noninstitutional representations, a sensitivity to contradiction, and, finally, the concepts of variable and of covariation between variables.
Rafael Nuñez's Special Lecture has been CANCELLED
Rafael Núñez

Rafael Núñez is Director of the Embodied Cognition Laboratory at University of California San Diego, USA. He investigates cognition from the perspective of the embodied mind. He si particularly interested in highlevel cognitive phenomena such as conceptual systems, abstraction, and inference mechanisms, and the biological and cultural phenomena that make them possible. His multidisciplinary interests bring him to address these issues from various interrelated perspectives: mathematical cognition, the empirical study of spontaneous gestures, cognitive linguistics, psychological experiments, neuroimaging, and field research with isolated groups such as the Aymara of the Andes' highlands and the Yupno of the remote mountains of Papua New Guinea (for a brief analysis of our studies on conceptions of time in these groups published in Science, see here: Aymara, Yupno). His book, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (with UC Berkeley linguist George Lakoff) proposed a new theoretical framework for understanding the human nature of Mathematics and its foundations. (Source: http://www.cogsci.ucsd.edu/~nunez/web/research.html) 
Special Tribute
In recognition and memory of Eugenio Filloy (19422020)
When planning the program for PMENA 42, we had envisioned, since 2019, including a special liveperson homage to Eugenio Filloy, cofounder in 1975 of the Mathematics Education Department of the Center for Advanced Studies and Research (Cinvestav) and a driving force of the Mathematics Education discipline at an international level from the early 1970s. Sadly, Eugenio Filloy passed away in March, 2020; but the more reason to include this special homage in his memory.