NCTM: News Bulletin: March 2000 : Cover Story 

Book's Messages Strike Chords In China, elementary school mathematics educators teach only mathematics. They teach just a few classes a day, with the rest of the day dedicated to planning and collaboration with colleagues. And they are generally much more agile with mathematics concepts than their American counterparts are, says a small comparative study of U.S. and Chinese elementary school teachers conducted in the late 1980s. This study forms the core of Liping Ma's book, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. The book is earning praise both from some of those who support changes proposed in the NCTM Standards and from some of those who oppose them, and it is sparking many discussions. It is also helping to unify some disparate forces in mathematics education on at least a few ideas for continuing positive changes (some of which have already begun in the decade since the research was done). Chinese elementary school teachers outperformed their American counterparts in a late 1980s comparison. Above are fourthgrade students working on enrichment activities in a mathematics elective class at Sanpailou Primary School, Nanjing, China. Photo by Frances R. Curcio. Studying American and Chinese Elementary School Teachers Ma came to her conclusions by analyzing the teachers' own problem solving on four identical questions. Ma, herself formerly an elementary school teacher in China and now with the Carnegie Foundation for the Advancement of Teaching, conducted the research for, and wrote the book as part of, her doctoral dissertation at the University of California at Berkeley. For the American portion, Ma used interview data from a late 1980s National Center for Research on Teacher Education (NCRTE) study of 23 American elementary school teachers. For the Chinese portion, she traveled to China to interview 72 Chinese teachers. The American teachers generally had five or six more years of formal schooling (a bachelor's degree in college and often one or two more years of formal study) than their Chinese counterparts. A number of the Americans who were studied were in a special master's degree program following their first full year of teaching (the Chinese teachers appeared to have more actual teaching experience than some of the Americans). Ma's comparison illustrates some skills differences between the American and Chinese teachers. Although the American elementary school teachers showed correct procedural knowledge of problems in wholenumber subtraction and multiplication, they were less skilled than the Chinese teachers in more advanced topics. Some were stumped by problems such as According to the book, "Only nine (43 percent) completed their computations and reached the correct answer." In contrast, the Chinese teachers studied struggled very little with the same problems. And in turn, they showed much more flexibility in challenging their students to learn. They were better able to pull from a wide variety of concept areasor "knowledge packages"to help students understand new ideas. Ma suggests that the Chinese performance is linked not only to good procedural understanding (which the American teachers displayed, but to a slightly lesser extent) but to something much deeper. She calls it "profound understanding of fundamental mathematics," which, she writes in the book, "goes beyond being able to compute correctly and give a rationale for computational algorithms. A teacher with profound understanding of fundamental mathematics is not only aware of the conceptual structure and basic attitudes of mathematics inherent in elementary mathematics, but is able to teach them to students." Ma found that 10 percent of Chinese teachers displayed such understanding, and 80 percent displayed at least some of that understanding. The Americans, however, appeared to Ma to be lacking in this area. Why the Difference? Deep mathematical understanding, she argues, may very well begin in the early years and then can be solidified through teaching experience. In the book, Ma compares the mathematics ability of Chinese ninth graders, Chinese preservice teachers, Chinese teachers, and the American teachers. The Americans lagged behind all
Besides their different school experiences, teachers from the two countries have much different professional experiences. Chinese elementary school mathematics teachers teach only one subject, allowing them to be mathematics specialists. Furthermore, they teach just a few 45minute classes (of up to 60 students each) a day. The rest of their day is spent collaborating with other teachers, grading papers, and planning lessons. One of the Chinese teachers highlighted in Ma's book notes, "I always spend more time on preparing [for] a class than on teaching, sometimes three, even four times, the latter." The time for professional growth the Chinese allow their teachers is the likely envy of many of their American counterparts, who often don't even have time for bathroom breaks, let alone discussion with their colleagues. The education system in China differs greatly. China has a centralized, state curriculum and just a small handful of textbooks from which to choose, in comparison with highly local decision making regarding curriculum in the United States and the plethora of teaching materials. Furthermore, the two countries' cultures differ vastly. Ma notes, "Cultural differences obviously play an important role. My belief is, however, that our teachers' math knowledge can be improved without changing their cultural background." The Two Systems Today A decade later, Ma's observations of Chinese elementary school teachers still seem to hold. Frances R. Curcio, a professor of mathematics education at New York University, led a delegation of U.S. teachers to China last summer. She observes, "[The Chinese teachers] were very knowledgeable of the underlying mathematics of the content they were presenting and reviewing, and the types of problems they selected. The depth of discussion during the lessons and during the postlesson discussions revealed their mathematical insights.... Typical of mathematics instruction in the lower grades, students [in the school she observed] solved puzzles and worked with manipulative materials such as tangrams." Curcio added, "The teachers pointed out that their salaries and jobs are dependent on students' good test scores." U.S. teaching, however, began to undergo some important changes since Ma conducted her research in the late 1980s. The NCTM Curriculum and Evaluation Standards was published in 1989, and an updated version, Principles and Standards for School Mathematics, will be published in April. Those documents both focus on improving mathematics throughout the grades, and Principles and Standards includes coverage of the prekindergarten years. Regarding teacher preparation, the National Council for Accreditation of Teacher Education is implementing more rigorous requirements for mathematics teacher educator programs in accredited institutions. NCTM itself is launching a professional development academy (see p. 2 of this News Bulletin) to help teachers teach mathematics better. Even politicians are joining the effort, with calls for more and betterprepared teachers and with the funding proposals to support the effort.
The book's messages about the need for elementary school mathematics teachers to have greater mathematical understanding resonates with many. The book confirms, and helps promote, the longunderstood notion among mathematics education researchers that, as Deborah Schifter, a senior scientist at the Education Development Center in Newton, Mass., says, "There is conceptual depth to the mathematics of the elementary grades."
This book appears to be most relevant to the preservice preparation of teachers, but its most powerful findings may well relate to our understanding of teachers' work and their careerlong professional development. Lee S. Shulman
At the teacher preparation level, the book suggests we need to do more to make sure prospective teachers have procedural and conceptual mathematics skills as well as skills in teaching. Says Curcio, "Mathematics teacher educators and mathematicians need to collaborate to develop ways to strengthen future teachers' understanding of underlying elementary mathematics. Simply having preservice teachers doing 'mathematics activities' without studying the underlying mathematics will continue to do a disservice to them and to their future students." Another consideration for preservice teacher education, notes Schifter, is "the importance of recognizing the mathematics in the contexts of the tasks of teaching. This also has implications for courses for teachers: that mathematics can be taught through examples of student thinking, analyses of activities from the K–6 curriculum, and writing or revising problems." Even beyond education, the book supports the need for, and indeed the educational benefits of, changing professional teaching conditions for U.S. teachers. "This book appears to be most relevant to the preservice preparation of teachers, but its most powerful findings may well relate to our understanding of teachers' work and their careerlong professional development," writes Lee S. Shulman of the Carnegie Foundation for the Advancement of Teaching in his foreword to Ma's book.
Liping Ma
Whether or not one agrees with the book, it provides some food for thought for everyone involved in improving mathematics education. And it supports the necessity, highlighted in NCTM's Standards documents, that even at the elementary school level, students can, and should, learn challenging mathematics. And teachersand what they knowmatter. For teachers, Ma's message is really quite straightforward: "Let's think seriously about the mathematics we are teaching, and learn more mathematics while we are teaching it." For More Information
Ma suggests that the Chinese performance is linked not only to good procedural understanding (which the American techers displayed, but to a slightly lesser extent) but to something much deeper.
three Chinese subgroups. From this comparison, it appears that Chinese students receive much more rigorous instructionand that it sticks with them. American education at the time this research was conducted tended to focus more on developing procedural knowledge and learning the "rules," and this was evident in the American teachers' approach to problem solving and teaching.
U.S. teaching . . . began to undergo some important changes since Ma conducted her research in the late 1980's.
To support constructive and purposeful changes, we will need to continue to change the public's perception of mathematics at all levels. Says HungHsi Wu, professor of mathematics at the University of California at Berkeley, "We need to spread the messageand NCTM can do this better than any other organization I knowthat learning mathematics requires sustained hard work and dedication, and that such effort is needed in order to bring the nation's math education to a higher level."
Carnegie Foundation for the Advancement of Teaching
At the university level, the book might be prompting a different look at the coursework and the measures of success. Says Roger Howe, a professor of mathematics at Yale University, "Most university mathematicians see much of advanced mathematics as a deepening and broadening, a refinement and clarification, an extension and fulfillment of elementary mathematics. However, it seems that it is possible to take and pass advanced courses without understanding how they illuminate more elementary material, particularly if one's understanding of that material is superficial." (From his review of Ma's book in the November 1999 Journal for Research in Mathematics Education).
Cultural differences obviously play an important role. My belief is, however, that our teachers' math knowledge can be improved without changing their cultural background.
The book's perspectives are uniting people who've long disagreed about the best directions for mathematics education. "Many worldclass mathematicians are rhapsodic about it ... That's because it says content knowledge makes a difference. But at the same time, those who have reform perspectivesthose who value a deep and connected view of mathematical thinking, and who understand that teacher competence includes having a rich knowledge base that contains a wide range of pedagogical content knowledgefind that the book offers riches regarding content, teacher preparation, and teacher professionalism," notes Alan Schoenfeld, a professor of mathematics education at the University of California at Berkeley.