Anne Broise: from mathematical abstraction to educational value

A native of Brittany, after her baccalaureate, Anne Broise opted for a 2-year preparatory course for a competitive exam. "I knew I liked mathematics, but I wasn't sure whether I wanted to be a teacher, researcher or engineer...," recalls the mathematician. So it wasn't until after her preparatory course and her admission to a chemistry school that she eventually decided to pursue an undergraduate degree in mathematics at the University of Western Brittany (UBO), at Brest. At this time, she began to take an interest in probability and ergodic theory, and went on to complete a DEA (Diploma of Advanced Studies, the equivalent of a 2nd year Master's degree in research today). She then went on to a thesis at the University of Rennes I, before being recruited as a lecturer at Université Paris-Sud (now Université Paris Saclay) in 1994.

A long-term commitment, a unique career path

Since then, Anne Broise has remained with the same institution. This choice runs counter to the mobility expected in the academic world, but is one for which she makes no excuses. "I didn't want to move, and I didn't want to take the Accreditation to Supervise Research (HDR). I had other, personal and family priorities, and I didn't see why I should give them up." Far from limiting her commitment, this loyalty to Orsay has led to a full career, at the crossroads of fundamental research and educating future teachers. A solid career path, impervious to the prevailing models, which illustrates another way of being a mathematician at the university.

Understanding chaos in simple systems

From her thesis onwards, Anne Broise tackled a fascinating question: how can simple mathematical rules, repeated over time, give rise to chaotic behaviour? She focused specifically on the dilating transformations of the interval that amplify the gaps between two neighbouring points. Her aim was to understand the distribution of orbits of these transformations, focusing on statistical results rather than the study of orbits of particular points. Anne Broise likes to use a simple example to illustrate her work: multiply a number between 0 and 1 by 10, then keep only the decimal part, and start again. "If you make a mistake on the thousandth digit after the decimal point, you think it won't matter. But after a thousand iterations, the two resulting numbers will bear no relation to each other."

Her thesis was structured around two key areas: refining theorems on the averaging behaviour of the orbits of these systems, and studying multidimensional continued fractions, in particular the Jacobi-Perron algorithm. "My central challenge concerned Lyapunov exponents, which measure how well rational approximations approach the starting point." After her thesis defence, she continued to study wide inequalities with Yves Guivarc'h, successfully demonstrating that all the inequalities obtained until then were in fact strict. "Our work validated the numerical calculations and produced a theoretical method for demonstrating inequalities between Lyapunov exponents for other algorithms," she says. These excellent results, published in 2000 in the mathematical journal Les Annales de l'Institut Fourier (The Annals of the Fourier Institute), marked a milestone in her scientific career.

New fields: geometry and trees

Subsequently, Anne Broise extended her research to new geometric contexts. Working with Frédéric Paulin, she continued to explore continued fractions, this time in characteristic p. "We were working in a different geometry, in a regular tree, and the question we asked ourselves was: can we establish, in this setting, the results that are already known on the interval?" By combining dynamics, coding and operators, they addressed counting problems, particularly those linked to the study of the geodesic flow in these particular spaces. This work resulted in several articles and a book.

Educating future teachers, a decisive turning point

Anne Broise's career took a decisive turn in the early 2000s, when she joined the team delivering the secondary school teaching certificate (CAPES - Certificat d'aptitude au professorat de l'enseignement du second degré) in mathematics. Previously fully committed to research, she discovered a field where she felt truly at home. "Students aren't always very good at maths, but that doesn't matter. The agreement is simple: to welcome them as they are and help them become the best teachers they can be." She quickly took on responsibilities within the MEEF Master's programme, designed to prepare secondary school teachers. She focuses on mathematics and assisting students with their work placements, dissertations and entry into the profession. "I had to develop the necessary skills to go and observe the trainee teachers in their classrooms and support them with their professional dissertations. I've learned to look at their classroom practices in a way that could be useful to them."

A view on knowledge transmission and equality in mathematics

Her commitment soon extended beyond the local context when she became the academic coordinator of the MEEF master's programme in maths for the entire Versailles academy. As a member of the Education Graduate School (EFE) at Université Paris-Saclay, she is involved in a broader discussion about teacher training and the development of the profession. She advocates a demanding, professionally focused programme, open to a wide range of profiles - including talented mathematicians who choose teaching rather than research. "We need to support these individuals in developing their own way of teaching and in gaining a perspective on both teaching methods and the maths they'll be expected to teach. A fundamental dimension of this curriculum is everything that comes with it, such as thinking about gender equality in education, learning how to address all students, including those from disadvantaged socio-professional backgrounds or those who don't like the subject. Mathematics teachers are currently struggling to make the subject less masculine and elitist. We're trying to help them with this."

When knowledge transmission becomes research

Her commitment also takes more experimental forms. In partnership with Mélanie Guenais, she teaches artistic workshops: mathematical theatre with the Terraquée company and paper folding with the paper folding organisation, the International Research Centre for Folding Modelling (CRIMP). These workshops are designed to strengthen the ability of future teachers to communicate and pass on mathematics in a different way. "We visited young teachers who were struggling to connect with their classes. We had to give them tools," she recalls. This project was initially funded by La Diagonale (the Arts, Culture, Science and Society department of Université Paris-Saclay) and then later by the Education Graduate School, the Scult interdisciplinary programme and a "Transform" call for projects from the University’s Educational Innovation department. It forms part of an action-research approach that explores how to "get all pupils doing mathematics", including through unexpected pathways. "Very clearly, the contribution of artistic practice during the training year enables young teachers to think about their classroom practice in new ways and enhance it." Today, Anne Broise is working on a publication, doing what she has always done: passing on her knowledge.