Profound Understand of Fundamental Whatever is the key to education reform. Elaborating a bit more on my plea for more educators to read the inspirational Liping Ma
... mathematics teaching in Chinese classrooms, even by a teacher with PUFM (Profound Understanding of Fundamental Mathematics) seems very "traditional"; that is, contrary to that advocated by reform. Mathematics teaching in China is clearly textbook based. In Chinese classrooms, students sit in rows facing the teacher, who is obviously the leader and maker of the agenda and direction in classroom learning. On the other hand, one can see in Chinese classrooms, particularly in those of teachers with PUFM, features advocated by reform - teaching for conceptual understanding, student's enthusiasm and opportunities to express their ideas, and their participation and contribution to their own learning processes. How can these seemingly contradictory features - some protested against and some advocated by reform - occur at the same time?
The perspective of Cobb and his colleagues helps to explain this puzzle ...
In their case study of two classrooms, one with a "tradition of school mathematics" where knowledge was "transmitted" from the teacher to "passive students" and one with a "tradition of inquiry mathematics" in which "mathematical learning was viewed as an interactive, constructive, problem centred process," the scholars found that in both the teachers and the students contributed to the development of their classroom mathematics tradition, while in both classrooms the teachers expressed their "institutionalised authority" during the process. Cobb and his associates suggested that "meaningful learning" may be mere rhetoric in mathematics education because "the activity of following procedural instructions can be meaningful for students" in certain classroom mathematics traditions. The transmission metaphor that describes traditional mathematics teaching as the attempt to transmit knowledge from the teacher to passive students may be appropriate only "in the political context of reform"
In this sense, although the mathematics teaching in Chinese teachers' classrooms does not meet some "rhetorical characterisations" of the reform, it is actually in the classroom mathematics tradition advocated by the current reform. In fact, even though the classroom of a Chinese teacher with PUFM may look very "traditional" in its form, it transcends the form in many aspects. It is textbook based but not confined to textbooks. The teacher is the leader, but students' ideas and initiatives are highly encouraged and valued.
On the other hand, from a teacher who cannot provide a mathematical explanation of algorithms for subtraction with regrouping, multidigit multiplication or division by fractions; from a teacher who cannot provide a correct representation for the meaning of an arithmetical operation such as division by fractions; or from a teacher who is not motivated to explore new mathematical claims, what kind of "teaching for understanding" can we expect?
Liping Ma, pp. 151-2