KNOWING AND TEACHING ELEMENTARY MATHEMATICS LIPING MA PDF

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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 riamemamohelp.cf At Right Here is where Liping Ma's book Knowing and.


Knowing And Teaching Elementary Mathematics Liping Ma Pdf

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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.

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. 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.

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? How can I reach these goals? In a word, one thing is to study whom you are teaching, the other thing is to study the knowledge you are teaching. If you can interweave the two things together nicely, you will succeed.

We think about these two things over and over in studying teaching materials. Believe me, it seems to be simple when I talk about it, but when you really do it, it is very complicated, subtle, and takes a lot of time. It is easy to be an elementary school teacher, but it is difficult to be a good elementary school teacher. We need many more commentaries on the teaching of mathematics like those contained in Mas book.

We also need more detailed lesson plans, as are frequently provided in Japan. However, these comments almost all deal with the initial steps of an idea, which typically means using pictures or manipulatives to try to get across the basic concept. Almost never is there elaboration of what should be done next, to help develop a deeper view of the subject, which will be necessary for later work. And elementar y school mathematics is much deeper, more profound, than almost everyone has thought it to be.

As Ma comments, toward the end of her book: In the United States, it is widely accepted that elementary mathematics is basic, superficial, and commonly understood. The data in this book explode this myth. I do not understand why it deserves that rating. I am quite familiar with this series, as I reviewed it as part of a textbook adoption process.

Regarding fractions, for example, Connected Math has some material on the addition and subtraction of fractions, but nothing as systematic as described by the Chinese teachers interviewed by Ma.

There is less on multiplication of fractions, and nothing on the division of fractions. If our students go through grade 8 without having studied the division of fractions, where are our future primary teachers going to learn this? Another recent development that leaves me less than encouraged is the way fractions are addressed in the draft of the revised K mathematics standards released last year by the National Council of Teachers of Mathematics Principles and Standards for School Mathematics: Discussion Draft 5.

Most of the work on fractions has been put in the grades 6 to 8 band. Students are to develop a deep understanding of rational number concepts and reasonable proficiency in rational-number computation. It is the adjective reasonable that bothers me. Proficiency should be the goal. It is hard to imagine the Chinese teachers that Ma interviewed settling for reasonable proficiency with fractions for their students. These lower expectations show in every international comparison.

Furthermore, the only problem used to illustrate division of fractions in NCTMs draft revision is how many pieces of ribbon yards long can be cut from 4 yards of ribbon. 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.

How would you approach these problems if you

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. 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.

Knowing & Teaching Elementary Mathematics.pdf

There is more we can do. 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.

Modern elementary mathematics

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. 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?Most of the work on fractions has been put in the grades 6 to 8 band.

He founded the Algebra Project. Within its pages, the highly thought-of social psychologist shows us how, even in the absence of explicit racism, negative stereotypes can continue to pervade American life, and have far-reaching influences on our behavior. 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?

The extreme traditionalist view is that knowledge should be didactically transmitted to obediently attentive pupils, who engage in closed-ended practice exercises and rote learning.

For example, computer modeling allows children to change parameters in virtual systems created by educators and observe emergent mathematical behaviors, or remix and create their own models.

Teachers manuals provide information about content and pedagogy, student thinking, and longitudinal coherence. Three levels of subtraction with regrouping problems are related to this central sequence: Page 15 Share Cite Suggested Citation:"Teachers' Understanding of Fundamental Mathematics.

MIKA from Rochester
Also read my other articles. I have always been a very creative person and find it relaxing to indulge in touch rugby. I am fond of reading comics sometimes .
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