The NCTM Principles and Standards for School Mathematics begins with the following vision of a “high-quality engaging mathematics instruction.”
Students confidently engage in complex mathematical tasks chosen carefully by teachers. They draw on knowledge from a wide variety of mathematical topics, sometimes approaching the same problem from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress. Teachers help students make, refine, and explore conjectures on the basis of evidence and use a variety of reasoning and proof techniques to confirm or disprove those conjectures. Students are flexible and resourceful problem solvers. Alone or in groups and with access to technology, they work productively and reflectively, with the skilled guidance of their teachers. Orally and in writing, students communicate their ideas and results effectively. They value mathematics and engage actively in learning it. (p. 3)
Such a vision is realized in many Hungarian classrooms, where, historically, a strong and explicit emphasis is placed on problem solving, mathematical creativity, and communication. In this Hungarian approach, 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.
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.
In such a learning environment, 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 mind used 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. Thus BSME prepares future teachers to address important national needs in mathematics education.
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.