Elementary Mathematics as Fundamental Mathematics. Profound How Liping Ma's Knowing and Teaching Mathematics Entered the U.S.. Mathematics and. This item:Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in by Liping Ma Paperback $ Keywords: PISA, Indian students, elementary school mathematics, teacher saicumspecsacont.gq At Right Here is where Liping Ma's book Knowing and.
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Library of Congress Cataloging-in-Publication Data Ma, Liping. Knowing and teaching elementary mathematics: teachers' understanding of fundamental. Subtraction, with or without regrouping, is a very early topic anyway. Is a deep understand- ing of mathematics necessary in order to teach it? Does such a. Liping Ma, Knowing and teaching elementary mathematics: Teachers' un- derstanding of fundamental mathematics in China and the United States,. Mahwah.
Less well known are formulas for solutions of cubic and quartic equations whose discovery was the high point of 16th century mathematics.
Their study forms the heart of this book, as part of the broader theme that a polynomial's coefficients can be used to obtain detailed information on its roots. The lack of a broad perspective is probably at least as big a problem today as it was then, so Klein's text is still valuable.
Klein also frequently discusses historical and pedagogical aspects, and the tone is quite informal throughout. Also available in English as ''Foundations of Analysis''.
AMS Chelsea Publishing. An axiomatic construction of numbers and concise derivation of their fundamental properties, from the natural numbers to the complex numbers. Lawrence Erlbaum Associates, Moses] and Charles E.
Cobb, Jr. Beacon, He has now returned to Mississippi to teach math to descendants of the sharecroppers he helped mobilize 40 years before. With journalist and fellow activist Cobb, he tells his personal story and shares his vision of universal math literacy among poor and minority children.
He founded the Algebra Project. Oxford University Press, 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.
Moreover, it may affect a teacher's goals for students—if knowing how is insufficient, students must come to understand why. In contrast, a teacher without this attitude may still know how and why, but not think it important that students know both.
Or a teacher without this attitude may know how, but not why—and may not be able to answer students' questions, nor see the importance of student questions. Thus a teacher's attitude may affect not only the mathematics the teacher knows but also the mathematics the teacher teaches.
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. Such attitudes may affect a teacher's knowledge by contributing to its coherence and connectedness—and also affect a teacher's teaching.
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.