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“The learning we want for our students, we must also want for ourselves” Jordi Deulofeu
Jordi Deulofeu, a mathematics teacher for over fifty years and a leading figure in math education and outreach, has trained generations of students and teachers alike.
Jordi, tell us about your career. I’ve been a math teacher for fifty years across various educational levels. I’ve taught high school and university, and I’ve supervised doctoral theses. I have also dedicated myself to educational outreach.
On the first day you walked into a classroom, were you already a good teacher? No! Good teaching is whatever generates good learning, and understanding how people learn is complicated. Back then, the CAP (Pedagogical Aptitude Course) was just a formality of little interest. So, at the beginning, I went into the classroom and “made the students suffer” by reproducing university models: just filling up chalkboards. I quickly realized that wasn’t working.
How did you turn things around? The key moment is reflecting on your practice: realizing you need to change. It was a turning point to join Grup Zero, with Carmen Escarate and Jaume Jorba—who was a sort of natural leader—along with Marta Berini, Carles Lladó, Maties… We were a group of secondary teachers from the UAB (Autonomous University of Barcelona) area who met on Tuesdays to think about how to improve our classes. That’s where I understood that science—and education—is not created by isolated individuals; it’s created by groups who think and work together.
From high school to university: what changes when you are the professor? There is a common thread: ensuring that students learn. But the method changes. In compulsory education (K-12), motivation is fundamental; if they don’t want to learn, they won’t.
You’ve supervised many doctoral theses. What do you take away from that experience? It has been very enriching. I’ve supervised about twenty-one, and they’ve all been different. When you see someone grow as a researcher and an educator, you realize that this mentorship is one of the most intense ways to practice teaching. The last thesis I supervised was Albert Vilalta’s!
About Math Education
It is often said that Catalonia has a powerful ecosystem for mathematics education. Where does this strength come from? It stems from the pedagogical tradition of the first third of the 20th century—Montessori, the Escuela Nueva (New School)—which was cut short by the Civil War but resurfaced in the sixties with the pedagogical renewal and movements like Rosa Sensat, and figures like Pere Puig Adam. During the Transition in the seventies and eighties, the creation of groups like Grup Zero or the Periòdica Pura group (Claudi Alsina) was key. These gave way to teacher associations federated under FEEMCAT. It’s similar to what Mersenne did in the 17th century by gathering mathematicians like Pascal or Descartes: science is made by groups, not individuals.
And when does “didactics” (math education) as a discipline come into play? Subject-specific didactics are necessary because knowing mathematics and knowing pedagogy separately isn’t enough. You have to know what math the students already know, what they are learning now, what they will learn later (the mathematical horizon), and what the curriculum says when broken down into “Big Ideas.” It’s a kind of “math for teaching” that is specific to the faculty. In Spain, this field was officially recognized in 1986. This has allowed for much more empirical research. Before, experts gave opinions; now, math education is based on evidence.
What does it mean to “do mathematics”? Paul Halmos said: “Mathematics is conjectures, properties, theorems, axioms… But the heart of mathematics, what makes math move forward, is problem-solving.” Therefore, doing math is solving problems using all the tools that constitute what we generally call mathematics.
I also like to quote Von Neumann: “In mathematics, you don’t understand things. You just get used to them. Having problems in math is the best thing that can happen to you because you can invent them. Having problems in life, on the other hand, isn’t exactly the best. If people think mathematics is difficult, it’s because they don’t realize how hard life is.”
What myths about learning math need to be debunked? First, the idea that “spare the rod, spoil the child” (or that learning must be painful). To me, that’s barbaric. If a student feels tortured, they might pass exams, but they won’t learn. You have to be able to enjoy mathematics. Not everyone will enjoy it at the same level, but those who don’t have a special inclination for math should at least have “mathematical success”—the experience of “I got this right.”
Second, the myth that math is just numbers, operations, and memorizing techniques. Those are necessary to solve problems, but they aren’t the essence. We should take the playful aspects of pleasurable learning more seriously… and perhaps not take math so seriously as a “hard subject.”
What makes a good math teacher? A teacher must know a lot of math. But which math? There is math specific to teaching: knowing what students know, what they are learning, what comes next, and how the curriculum is organized into big ideas. But that’s not enough. A fundamental point is classroom management. The classroom is a high-tension, complex environment where many things happen very quickly. Managing that complexity and knowing how to make real-time decisions is key. A good teacher is one who knows how to “stretch” their students and maintains a certain tension in the room so that everyone is working and moving forward, each at their own level.
What should teacher training look like? Considering that the Master’s for Secondary Education is short, if I had to ask for just one goal, it would be this: that future teachers understand the profession is very complex and they will need to be lifelong learners. If a teacher leaves the Master’s thinking they know enough, we’re lost.
We also need to find a better relationship between theory and practice. And a serious problem in this country is entry into the profession: it shouldn’t be that new teachers are often placed in the most difficult situations without support. That burns out teachers who could have been great.
In an ideal math classroom, what should happen? In any class, there must be learning. For that to happen, there must first be a problem or a question. Then, thinking and reasoning, and interaction: we must be able to discuss and contrast ideas. Finally, you have to “institutionalize” what has been done (summarize the formal concepts learned). The activity should be what we call “low floor, high ceiling”: every student can start doing something, but the activity can be stretched as far as the most capable students can go.
Are we far from that? What do you think of the “competencies vs. knowledge” debate? I consider this debate absolutely sterile. You cannot develop competencies without having the necessary knowledge. But focusing only on knowledge, thinking they will “apply the skills later,” works even less. Furthermore, you have to add the practice of routines. But not “reproductive practice” (rote drill); rather, what we call “productive practice”: the kind that allows you to practice while working on other competencies and goals.
What structural changes would you like to see in the system? Above all, changes that support teacher improvement. For example, curricular stability. You have to let professionals work for a certain amount of time. Then, connect the research–teacher–administration triangle. Research must listen to the real problems of teachers, and the administration must make educational policy decisions, not just political decisions. There must be a real transfer between decision-makers and the reality of the classroom.
What are the keys to getting research into the classroom? On one hand, research should study problems that actually affect the classroom. On the other, it should provide the building blocks to create projects that can be implemented. Over the years, there have been magnificent projects (like Grup Zero or Paolo Boero’s in Italy), but few have managed to reach a large number of schools. That is what Innovamat is achieving: getting many schools to use research-based resources to teach math in their classrooms.
And what role does outreach play? Enormous! I participated in the “white paper” of outreach in Catalonia because we need to offer the world an image of mathematics that is closer to what it actually is, and also show that math is important within the landscape of knowledge. In fact, my current experience teaching courses at the Aules de la Gent Gran (Senior Citizens’ University) is very revealing. Many students say: “If math had been explained to me like this, I might have liked it.”
Games and History
Jordi, you love to play. What is the relationship between games, math, and learning? A game, understood as a playful activity, has enjoyment as its primary component. And to learn, you must be able to enjoy yourself. Most games have a connection to math—even if it’s just counting points—but the deep connection is that in a game, we are solving a problem, reasoning, and designing strategies.
Why is play a good context for learning in the classroom? Because it shares key characteristics with learning: freedom, for example, since playing is a free choice, just like wanting to learn. Or rules, because in a game there are rules that must be followed, just like in math. Even the pursuit of a goal: in a game, you want to win; in math, you want to solve the problem and argue that the solution is valid.
As a learning context, do you think it’s important to include the History of Mathematics in the classroom? Of course. When we talk about learning contexts, it’s often oversimplified as “math and the real world.” That’s not practical, because math is already part of the real world. I like the historical context because it shows math as a cultural activity that has always existed, and because many school problems have a history and an original utility—like indirect measurement—that are hard to explain conceptually today with modern technology like lasers.
Can you recommend some games from your “pantry”? Sure. As a fan and member of the Set de Mates group—where we test games constantly—I recommend a few linked to concepts. For numbers, Trio, a new card game that is very beautiful and very “mathematical.” For probability, Can’t Stop, an addictive dice game that beats the typical school “camel race” activity. For combinatorics and speed, Set, a classic for making combinations and applying criteria. For geometry, Patchwork or the classic Ubongo. And for reasoning and scientific thinking, Eleusis. It might not fit in the classroom because games can last five or six hours, but it’s the great game of induction. It simulates the creation of the universe and helps you understand what it means to conjecture: moving from the specific to the general.
Final Messages for Teachers
What was the “class of your life”? I remember three especially. At the Costa i Llobera school, when I “flipped the switch” and told the students they were the ones who had to work. We turned the classroom into a camera obscura to project the sun, make triangles, and calculate distances. I felt like I was learning with them.
Also, the experience at Els Pins del Vallès school, in collaboration with the UAB: I taught math to 8th graders (current 8th grade/2nd year of ESO) and my students from the Faculty of Education would come to observe, and then we would reflect together. We worked alongside Neus Sanmartí, who taught science.
And finally, my last class. It was emotionally very significant, organized by friends and students. In that moment, you see a whole life dedicated to teaching summarized.
What would you say to a teacher who has just started? That being a teacher is the most beautiful profession in the world because seeing how students learn is fantastic, but at the same time, it is very complex. It follows that you have to be learning your whole life. The learning we want for our students, we must also want for ourselves.
And to a teacher who has lost hope or is disappointed? I would say: “Keep watching how your students learn.” It’s true that circumstances have changed and everything is more complex now, but a teacher with years of experience has enough capacity to adapt to the times. Above all, keep looking for those learning opportunities in your students, because they are there.
