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This book describes the nature and development of the "profound understanding of fundamental mathematics" that elementary teachers need to become accomplished mathematics teachers, and suggests why such teaching knowledge is much more common in China than in the United States, despite the fact that Chinese teachers have less formal education than their U.
This book describes the nature and development of the "profound understanding of fundamental mathematics" that elementary teachers need to become accomplished mathematics teachers, and suggests why such teaching knowledge is much more common in China than in the United States, despite the fact that Chinese teachers have less formal education than their U. The studies described in this volume suggest that Chinese teachers begin their teaching careers with a better understanding of elementary mathematics than mat of most U. Their understanding of the mathematics they teach and-equally important-of the ways that elementary mathematics can be presented to students, continues to grow throughout their professional lives. Teaching conditions in the United States, unlike those in China, militate against the development of elementary teachers' mathematical knowledge and its organization for teaching. The concluding chapter of the book suggests changes in teacher preparation, teacher support, and mathematics education research that might allow teachers in the United States to attain profound understanding of fundamental mathematics.
A short summary of this paper. How special? Well, just how many people do you know whose work has had such impact that it became the subject of a study by the National Academy of Sciences? See the article by Fang and Paine reproduced at the end of this volume. Fang and Paine provide some detail on Liping's background, so I won't repeat the details of her history.
Suffice it to say that it was a great pleasure having Liping join my research group with the intention of converting her dissertation into a book, connecting her with Cathy Kessel, and working with the two of them to produce the final manuscript. Liping is passionate about her work, which she knows can make a difference. She brings to it the same sense of wisdom and determination that enabled her to survive her daunting responsibilities during the Cultural Revolution in China, her emigration to the United States, and her progress to the Ph.
But why has the book been so well received? In my opinion, Knowing and Teaching Elementary Mathematics has the delightful property that everyone who cares about mathematics teaching and learning will find something in the book that resonates deeply with their understandings-and, will learn from the book. I have roots in both the mathematical and educational communities I began my career as a research mathematician, but turned my attention to research in mathematics education many years ago , so let me say why this book appeals to both the mathematician and the educator in me.
Speaking as a mathematician, I'll simply say that it rings true mathematically. When I first read Liping's work, my reaction was, "she knows what she's talking about. She focuses on important mathematical ideas, and she gets them right. These are the kinds of understandings I want students to develop about mathematics. In Liping's world, and in mine, the mathematical operations we perform make sense. Whether it's the subtraction algorithm, which is grounded in an understanding of base arithmetic, or division by fractions, which can be understood in many different ways, the point is that math isn't arbitrary.
It isn't made up of random rules, such as "ours is not to reason why; just invert and multiply. I resonate strongly to that, and so, I suspect, do many of the mathematicians e. Reading it allows you to feel mathematical coherence. This is new to some, and resonates with others.
Speaking as a mathematics educator, I have exactly the same feelings of resonance. The mathematician in me knows lots about fractions. I can describe them as equivalence classes of ordered pairs, I can talk about the rational numbers being dense in the real numbers, and more. In fact, that's the way mathematicians think: there's lots of mathematics related to fractions, as we go up the ladder of abstraction. But Liping's discussions pull us in another I shall try to clarify these somewhat cryptic observations in this foreword, but first a brief biographical note about Liping Ma.
Liping became an elementary school teacher courtesy of China's Cultural Revolution. An eighth-grade middle-school student in Shanghai, she was sent to "the countryside"-in her case a poor rural village in the mountainous area of South China-to be re-educated by the peasants working in the fields.
After a few months, the village head asked Liping to become a teacher at the village school. As she has described it to me, she was a Shanghai teenager with but eight years of formal education struggling to teach all the subjects to two classes of kids in one classroom. Over the next seven years, she taught all five grades and became principal of the school.
A few years later, she would be hired as the Elementary School Superintendent for the entire county. When she returned to Shanghai filled with curiosity about her new calling, she found a mentor in Professor Liu, who directed her reading of many of the classics of educationamong them Confucius and Plato, Locke and Rousseau, Piaget, Vygotsky, and Bruner.
She longed to study even more, and to pursue her further education in the United States. IntroductionChinese students typically outperform U. Paradoxically, Chinese teachers seem far less mathematically educated than U.
Most Chinese teachers have had 11 to 12 years of schooling-they complete ninth grade and attend normal school for two or three years. In contrast, most U. In this book I suggest an explanation for the paradox, at least at the elementary school level. My data suggest that Chinese teachers begin their teaching careers with a better understanding of elementary mathematics than that of most U. Their understanding of the mathematics they teach and-equally important-of the ways that elementary mathematics can be presented to students continues to grow throughout their professional lives.
I document the differences between Chinese and U. I also document some of the factors that support the growth of Chinese teachers' mathematical knowledge and I suggest why at present it seems difficult, if not impossible, for elementary teachers in the United States to develop a deep understanding of the mathematics they teach. I shall begin with some examples that motivated the study.
In , I was a graduate student at Michigan State University. To make this meaningful for kids, something that many teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story-problems to show the application of some particular piece of content.
What would you say would be a good story or model for? I was particularly struck by the answers to this question. Very few teachers gave a correct response. Most, more than preservice, new, and experienced teachers, made up a story that represented , or. Many other teachers were not able to make up a story. The interviews reminded me of how I learned division by fractions as an elementary student in Shanghai. My teacher helped us understand the relationship between division by fractions and division by positive integers-division remains the inverse of multiplication, but meanings of division by fractions extend meanings of whole-number division:xxiv Introduction the measurement model finding how many halves there are in and the partitive model finding a number such that half of it is.
The understanding of division by fractions shown by my elementary school teacher was typical of my colleagues. How was it then that so many teachers in the United States failed to show this understanding? Several weeks after I coded the interviews, I visited an elementary school with a reputation for high-quality teaching that served a prosperous White suburb. With a teacher-educator and an experienced teacher, I observed a mathematics class when a student teacher was teaching fourth graders about measurement.
During the class, which went smoothly, I was struck by another incident. After teaching measurements and their conversions, the teacher asked a student to measure one side of the classroom with a yardstick. The student reported that it was 7 yards and 5 inches. He then worked on his calculator and added, "7 yards and 5 inches equals 89 inches.
The apparent mismatch of the two lengths, "7 yards and 5 inches" and "89 inches," seemed conspicuous on the chalkboard. It was obvious, but not surprising, that the student had misused conversion between feet and inches in calculating the number of inches in a yard. What surprised me, however, was that the apparent mismatch remained on the chalkboard until the end of the class without any discussion.
What surprised me even more was that the mistake was never revealed or corrected, nor even mentioned after the class in a discussion of the student teacher's teaching. Neither the cooperating teacher nor the teacher-educator who was supervising the student teacher even noticed the mistake.
As an elementary teacher and as a researcher who worked with teachers for many years, I had developed certain expectations about elementary teachers' knowledge of mathematics. However, the expectations I had developed in China did not seem to hold in the United States.
The more I saw of elementary mathematics teaching and research in the United States, the more intrigued I became. Even expert teachers, experienced teachers who were mathematically confident, and teachers who actively participated in current mathematics teaching reform did not seem to have a thorough knowledge of the mathematics taught in elementary school. Apparently, the two incidents that had amazed me were only two more examples of an already widespread and well-documented phenomenon.
The study measured achievement in various mathematical topics in each of 12 different countries at Grades 8 and In the early s, IEA carried out another study. The Second International Mathematics Study compared 17 countries in the Grade 8 component and 12 in the Grade 12 component.
Could it be that the "learning gap" was not limited to students? If so, there would be another explanation for U. Unlike factors outside of classroom teaching, teachers' knowledge might directly affect mathematics teaching and learning. Moreover, it might be easier to change than cultural factors, such as the number-word s ystem 7 or ways of raising children. It seemed strange that Chinese elementary teachers might have a better understanding of mathematics than their U.
Chinese teachers do not even complete high school; instead, after ninth grade they receive two or three more years of schooling in normal schools. However, I suspected that elementary teachers in the two countries possess differently structured bodies of mathematical knowledge, that aside from subject matter knowledge "equal to that of his or her lay colleague" Shulman, , a teacher may have another kind of subject matter knowledge.
For example, my elementary teacher's knowledge of the two models of division may not be common among high school or college teachers. This kind of knowledge of school mathematics may contribute significantly to what Shulman called pedagogical content knowledge-"the ways of representing and formulating the subject that make it comprehensible to others" p.
I decided to investigate my suspicion. Comparative research allows us to see different things-and sometimes to see things differently. My research did not focus on judging the knowledge of the teachers in two countries, but on finding examples of teachers' sufficient subject matter knowledge of mathematics.
Such examples might stimulate further efforts to search for sufficient knowledge among U. Moreover, knowledge from teachers rather than from conceptual frameworks might be "closer" to teachers and easier for them to understand and accept. Two years later, I completed the research described in this book.
encountered of teachers explaining what it means to really know and be able to teach elementary school mathematics. As the word “understanding” continues.
This is a preview of subscription content, access via your institution. Rent this article via DeepDyve. Ball, D. Google Scholar. Simon, M.
Not a MyNAP member yet? Register for a free account to start saving and receiving special member only perks. Following the small-group discussions on the preworkshop assignment from Liping Ma's book Knowing and Teaching Elementary Mathematics , a panel set the stage for thinking about the two overarching questions that were the focus of the workshop:. What mathematical knowledge do elementary teachers use in teaching? Teachers were asked how they would respond to classroom scenarios in which mathematical ideas played crucial roles.
The 20th anniversary edition of this groundbreaking and bestselling volume offers powerful examples of the mathematics that can develop the thinking of elementary school children. Studies of teachers in the U. Yet, these studies give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by reforms in mathematics education.
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels.
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Пожав плечами, он подошел к раковине. Раковина была очень грязной, но вода оказалась холодной, и это было приятно. Плеснув водой в глаза, Беккер ощутил, как стягиваются поры. Боль стала утихать, туман перед глазами постепенно таял.
Да будь они прокляты, эти криптографы. Ничего не понимают в системах безопасности. Присяга, которую Чатрукьян принимал, поступая на службу в АНБ, стала непроизвольно прокручиваться в его голове. Он поклялся применять все свои знания, весь опыт, всю интуицию для защиты компьютеров агентства, стоивших не один миллион долларов. - Интуиция? - с вызовом проговорил .
Мидж продолжала читать. Мгновение спустя она удовлетворенно вскрикнула: - Я так и знала. Он это сделал. Идиот! - Она замахала бумагой. - Он обошел Сквозь строй. Посмотри.
Да нет же, черт возьми. И кто только распустил этот слух. Тело Колумба покоится здесь, в Испании. Вы ведь, кажется, сказали, что учились в университете. Беккер пожал плечами: - Наверное, в тот день я прогулял лекцию. - Испанская церковь гордится тем, что ей принадлежат его останки. Испанская церковь.
Она посмотрела на часы, потом на Стратмора. - Все еще не взломан. Через пятнадцать с лишним часов. Стратмор подался вперед и повернул к Сьюзан монитор компьютера. На черном поле светилось небольшое желтое окно, на котором виднелись две строчки: ВРЕМЯ ПОИСКА: 15:09:33 ИСКОМЫЙ ШИФР: Сьюзан недоуменно смотрела на экран.
Однако в списке было еще одно сообщение, которого он пока не видел и которое никогда не смог бы объяснить. Дрожащей рукой он дал команду вывести на экран последнее сообщение. ОБЪЕКТ: ДЭВИД БЕККЕР - ЛИКВИДИРОВАН Коммандер опустил голову. Его мечте не суждено сбыться.
- На экране появилось новое окошко. - Хейл - это… Сьюзан замерла. Должно быть, это какая-то ошибка.
А потом медленно скрылась из виду в клубах пара. - Отчаянный парень, - пробормотал Хейл себе под нос. Он знал, что задумал Чатрукьян. Отключение ТРАНСТЕКСТА было логичным шагом в случае возникновения чрезвычайной ситуации, а ведь тот был уверен, что в машину проник вирус.
О. Я вижу, вам действительно очень нужно это Кольцова. Беккер мрачно кивнул. - Кому вы его продали.
Мы должны вырубить питание главного банка данных. - Это невозможно, - сказал директор. - Вы представляете, каковы будут последствия. Джабба отлично знал, что директор прав. Более трех тысяч узлов Независимой цифровой сети связывают весь мир с базой данных агентства.
Так это клипса. - Да, - сказала девушка.
Я вас знаю. На такой риск вы не пойдете. Сьюзан было запротестовала, но Стратмор не дал ей говорить.
На экране высветилось: СЛЕДОПЫТ ОТПРАВЛЕН Теперь надо ждать. Сьюзан вздохнула. Она чувствовала себя виноватой из-за того, что так резко говорила с коммандером. Ведь если кто и может справиться с возникшей опасностью, да еще без посторонней помощи, так это Тревор Стратмор.
И, полагаю, если с Танкадо что-нибудь случится, эта загадочная личность продаст ключ. - Хуже. Если Танкадо убьют, этот человек опубликует пароль.
My data suggest that Chinese teachers begin their teaching careers with a better understanding of elementary mathematics than that of most U.S. elementary.Arinersper 16.12.2020 at 23:46
Author's Preface to the Anniversary Edition Series Editor's Introduction to the Anniversary Edition A Note about the Anniversary Edition Foreword.