# KNOWING AND TEACHING ELEMENTARY MATHEMATICS LIPING MA PDF

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 yazik.info 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.

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. 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.

The text continues with: The image is of repeatedly cutting off of a yard of ribbon. Having students work with concrete objects or drawings is helpful as students develop and deepen their understanding of operations. It seems that we are back again to simple fractions and concrete objects that students can visualize.

Contrast this with what Liping Ma observed: The concept of fractions as well as the operations with fractions taught in China and the U.

Although Chinese teachers also use these shapes when they introduce the concept of a fraction, when they teach operations with fractions they tend to use abstract and invisible wholes e.

Here is part of her description: A teacher with PUFM is aware of the simple but powerful basic ideas of mathematics and tends to revisit and reinforce them.

He or she has a fundamental understanding of the whole elementary mathematics curriculum, thus is ready to exploit an opportunity to review concepts that students have previously studied or to lay the groundwork for a concept to be studied later.

However, PUFM did not come directly from their studies in school, but from the work they did as teachers. These teachers did not specialize in mathematics in normal school, which is what their teacher preparation schools are called.

But after they started teaching, most of them taught only mathematics or mathematics and one other subject. This allowed them to specialize in ways that few of our eleFALL mentary school teachers can.

## Knowing & Teaching Elementary Mathematics.pdf

Quite a few regularly changed the level at which they taught. They might go through a cycle of three grades, then repeat the same cycle, or change and teach a different age group. This allows them to see the development of mathematics from the perspective of a teacher, something too few of our elementary school teachers are able to do. Recently, the Learning First Alliancean organization composed of many of the major national education organizationsrecommended that beginning in the fifth grade, every student should be taught by a mathematics specialist.

There is more we can do.

## Modern elementary mathematics

Our teachers need good textbooks. They need much better teachers manuals. As noted before, our college math courses for future teachers at all levels need to be improved.

And just ask any teacher who has sat through mindless workshops whether our in-service professional development isnt long overdue for major overhaul. Teachers also need time to prepare their lessons and further their study of mathematics. Recall Mas comments that it is during their teaching careers that Chinese teachers perfect their knowledge of mathematics. Listen to this Shanghai teacher describe his class preparation: I always spend more time on preparing a class than on teaching, sometimes three, even four, times the latter.

## The basis of it all

I spend the time in studying the teaching materials: What is it that I am going to teach in this lesson? How should I introduce the topic?

What concepts or skills have the students learned that I should draw on? Is it a key piece on which other pieces of knowledge will build, or is it built on other knowledge?

If it is a key piece of knowledge, how can I teach it so students can grasp it solidly enough to support their later learning? If it is not a key piece, what is the concept or the procedure it is built on? How am I going to pull out that knowledge and make sure my students are aware of it and the relation between the old knowledge and the new topic? What kind of review will my students need? How should I present the topic step-by-step?

How will students respond after I raise a certain question? Where should I explain it at length, and where should I leave it to students to learn it by themselves?

What are the topics that the students will learn which are built directly or indirectly on this topic? How can my lesson set a basis for their learning of the next topic, and for related topics that they will learn in their future? What do I expect the advanced students to learn from the lesson? What do I expect the slow students to learn?The ultra-modernist believes that pupils should learn at their own individual pace, and partly by investigative activities, with the teacher being the facilitator of such a process.

The text continues with: The image is of repeatedly cutting off of a yard of ribbon. He was, first and foremost, a mathematician who constructed the best psychological analysis of what it means to understand mathematics, and his book encapsulates and expands upon all of the ideas propounded by Ma.

How should I present the topic step-by-step?

Even worse, conditions in the United States militate against the development of elementary teachers mathematical knowledge What are the topics that the students will learn which are built directly or indirectly on this topic?

Profound has three related meanings —deep, vast, and thorough—and profound understanding reflects all three.

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How am I going to pull out that knowledge and make sure my students are aware of it and the relation between the old knowledge and the new topic? And just ask any teacher who has sat through mindless workshops whether our in-service professional development isnt long overdue for major overhaul.

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