Teachers were asked how they would respond to classroom scenarios in which mathematical ideas played crucial roles. Her memories of elementary teaching in China suggested that Chinese teachers would react quite differently. She investigated this suspicion in her dissertation research, asking Chinese elementary teachers the same questions that had been asked of U. To analyze the responses of both groups of teachers, she developed the notion of profound understanding of fundamental mathematics PUFM.

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Teachers were asked how they would respond to classroom scenarios in which mathematical ideas played crucial roles. Her memories of elementary teaching in China suggested that Chinese teachers would react quite differently.

She investigated this suspicion in her dissertation research, asking Chinese elementary teachers the same questions that had been asked of U.

To analyze the responses of both groups of teachers, she developed the notion of profound understanding of fundamental mathematics PUFM. Fundamental mathematics is a foundation for later learning. Profound has three related meanings —deep, vast, and thorough—and profound understanding reflects all three. A deep understanding of fundamental mathematics is defined to be one that connects topics with ideas of greater conceptual power.

A broad understanding connects topics of similar conceptual power. And thoroughness is the capacity to weave all parts of the subject into a coherent whole. Profound understanding of fundamental mathematics is an understanding of fundamental mathematics that is deep, broad, and thorough. Teachers with PUFM are able to reveal and represent ideas and connections in terms of mathematics teaching and learning.

Such teaching and learning tends to be connected, display multiple perspectives, demonstrate awareness of basic ideas of mathematics, and have longitudinal coherence.

Like a taxi driver who knows a road system well, teachers with PUFM know many connections among past, present, and future under- 1 The essence of this paper was contained in the remarks of Liping Ma at the opening panel session of the Workshop.

They know how to guide students from their current understandings to further learning and to prepare them for future travel. Such teaching and learning is possible because the road system of fundamental mathematics has depth, breadth, and thoroughness, allowing teachers to connect student understandings with topics to be learned.

This is not the case in the United States, where knowing elementary mathematics is sometimes, perhaps often, construed as knowing how to add, subtract, multiply, and divide whole numbers and fractions. In contrast, a teacher without this attitude may still know how and why, but not think it important that students know both. Other mathematical attitudes displayed by Chinese teachers include the following: claims must be justified with mathematical arguments, it is desirable to approach the same topic in multiple ways, and it is desirable to preserve the consistency of an idea in different contexts.

These fall in the category of what Jerome Bruner calls basic attitudes and considers as one aspect of the structure of a discipline. Another aspect of disciplinary knowledge identified by Bruner is basic principles. In the case of elementary mathematics and perhaps all disciplines , basic attitudes have a symbiotic relationship with basic principles.

For example, justifications in elementary mathematics often draw on the distributive law. Solving a fraction problem in multiple ways might draw on relationships between a fraction and a division, division as the inverse of multiplication, or relationships between fractions and decimals.

In the base system, noting the consistency of the relationship between 10 and 1, and 10, and so on leads to the idea of the rate of Each unit of higher value is composed of 10 or powers of 10 lower value units. This leads to the more general principle of the rate of composing a higher valued unit—the rate is 10 in the base system, but there are other possibilities.

For instance, the binary system has a rate of 2. Like basic attitudes, basic principles may play a role in teaching, as well as knowing, mathematics. And you should know the role of the present knowledge in that package. You have to know that the knowledge you are teaching is supported by which ideas or procedures, so your teaching is going to rely on, reinforce, and elaborate the learning of these ideas.

Ma, , p. When U. The mathematical concept and the computational skill of multidigit multiplication are both introduced in the learning of the operation with two-digit numbers.

So the problem may happen and should be solved at that stage. Ma, , pp. Some are considered key pieces, and teachers take particular care that students understand them. Such attention to an idea in its first and simplest form allows teachers to pay less attention to later and more complicated forms. Rather, I let them learn it [on] their own. Instead, one needs to decompose the 1 as 10 ones and group some or all of the 10 ones with the 5 e.

Key pieces of the package have thick borders. The central sequence in the subtraction package goes from the topic of addition and subtraction within 10, to addition and subtraction within 20, to subtraction with regrouping of numbers between 20 and , then to subtraction of large numbers with regrouping.

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She explains the reasons behind differences in student comprehension and performance both on standardized tests and in life-long mathematical achievement. These reasons boil down to a primarily procedural understanding and explanation of elementary maths procedures, many of which are flawed in addition to their fundamental inadequacy. A deeper reason for the methods is the lack of knowledge inculcated in US teachers with regard to what Ma calls the "Profound Understanding of Fundamental Mathematics. It is as essential as learning and understanding the alphabet and phonics.

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