*By Ryota Matsuura, Assistant Professor of Mathematics at St. Olaf College and North American Director of Budapest Semesters in Mathematics Education.*

Home to eminent mathematicians such as Paul Erdős, John von Neumann, and George Pólya, Hungary has a long tradition of excellence in mathematics education. In the *Hungarian approach* to learning and teaching, a strong and explicit emphasis is placed on problem solving, mathematical creativity, and communication. Students learn concepts by working on problems with complexity and structure that promote perseverance and deep reflection. These mathematically meaningful problems emphasize procedural fluency, conceptual understanding, logical thinking, and connections between various topics.

For each lesson, a teacher selects problems that embody the mathematical goals of the lesson and provide students with opportunities to struggle productively towards understanding. The teacher carefully sequences the problems to provide focus and coherence to the lesson. These problems do more than provide students with opportunities to learn the mathematical topics of a given lesson. Indeed, the teacher sees the problems she poses as vehicles for fostering students’ reasoning skills, problem solving, and proof writing, just to name a few. An overarching goal of every lesson is for students to learn what it means to engage in mathematics and to feel the excitement of mathematical discovery. Click here for a sample task from a 5th grade classroom at Fazekas Mihály School in Budapest.

Another hallmark of the Hungarian approach is the classwide discussion of approaches to problems. After working on problems individually or in small groups, volunteers come to the front of class to share their solutions. Because of the non-trivial nature of these problems, students learn to communicate their thinking with clarity and precision. When a student gets stuck, others chime in to offer support and suggestions in a friendly manner. The teacher creates a welcoming environment that is conducive to the sharing of students’ mathematical experiences.

In such a classroom, the teacher’s role is that of a motivator and facilitator. He provides encouragement and support as students engage with the task at hand. He offers guidance when a student is stuck and probes when clarification is needed. After the student investigation, the teacher highlights important ideas embedded in a concrete problem, and summarizes and generalizes their findings. In particular, the teacher’s summary makes sense and is meaningful, because students have had the experience of playing around with these ideas on their own before coming together to formalize them as a class.

Moreover, the teacher repeatedly asks, “Did anyone get a different answer?” or “Did anyone use a different method?” to elicit multiple solutions strategies. This highlights the connections between different problems, concepts, and areas of mathematics and helps develop students’ mathematical creativity. Creativity is further fostered through acknowledging “good mistakes.” Students who make an error are often commended for the progress they made and how their work contributed to the discussion and to the collective understanding of the class.

My interest in this approach to teaching has led to my involvement in Budapest Semesters in Mathematics Education (BSME), a new semester-long program in Budapest, Hungary designed to introduce the Hungarian approach to American and Canadian undergraduates and recent graduates. Conceived by the founders of Budapest Semesters in Mathematics (BSM), BSME is specifically intended for students who are not only passionate about mathematics, but also the teaching of mathematics. Participants will immerse themselves in mathematical exploration to experience first-hand learning in the Hungarian approach; then they will investigate how to bring this pedagogy into their own future classrooms. They will observe Hungarian mathematics classrooms and will have the opportunity to plan and teach their own lessons to Hungarian students (in English).

One of the core benefits of the Hungarian approach, and one that I am excited for BSME participants to bring back to the US, is that students acquire the mathematical habits of mind that allow them to think like a mathematician. As Cuoco, Goldenberg, and Mark describe,

Much more important than specific mathematical results are the

habits of mindused by the people who create those results. … The goal is not to train large numbers of high school students to be university mathematicians. Rather, it is to help high school students learn and adopt some of the ways that mathematicians think about problems. … Although it is necessary to infuse courses and curricula with modern content, what is even more important is to give students the tools they will need in order to use, understand, and even make mathematics that does not yet exist.(pp. 375-376)

Given the wide-spread adoption of the Common Core State Standards, as well as the recently published *Mathematical Education of Teachers II* (MET2) report by CBMS and NCTM’s *Principles to Actions*, our teachers are now expected to provide learning experiences that lead to the acquisition and development of students’ mathematical habits of mind, so that “all students learn to become mathematical thinkers and are prepared for any academic career or professional path that they choose.”* Preparing teachers with the knowledge and skill set to cultivate such a learning environment is now an important national need in mathematics education, and the Hungarian approach has the potential to play a critical role in this endeavor.

To learn more about the Hungarian approach, consult the following articles:

- Andrews, P., & Hatch, G. (2001). Hungary and its characteristic pedagogical flow.
*Proceedings of the British Congress of Mathematics Education*, 21(2). 26-40. - Stockton, J. C. (2010). Education of Mathematically Talented Students in Hungary.
*Journal of Mathematics Education at Teachers College*, 1(2), 1-6.

*National Council of Teachers of Mathematics. (2014). *Principles to Actions: Ensuring Mathematics Success for All*. Reston, VA: Author, p. vii.

The Proceedings of the British Congress of Mathematics Education article has moved to:

http://www.bsrlm.org.uk/wp-content/uploads/2016/02/BSRLM-IP-21-2-8.pdf