They also lead the way into the more structured and even more creative work of noticing, remixing and building mathematical patterns. Far better, she says, to start by creating rich and social mathematical experiences that are complex allowing them to be taken in many different directions yet easy making them conducive to immediate play.
At the free play level, you are learning in a very fundamental way—you really own your concept, mentally, physically, emotionally, culturally.
What is learned without play is qualitatively different. It helps with test taking and mundane exercises, but it does nothing for logical thinking and problem solving. Then comes the formal level, where students can use abstract words, graphs, and formulas. But ideally, a playful aspect is retained along the entire journey. People are different, and people need to approach mathematics differently.
For example, in a group learning about the properties of rhombuses, an artistically inclined person might prefer to draw a rhombus, a programmer might code one, a philosopher might discuss the essence of rhombi, and an origami master might fold a paper rhombus. Many people live to a ripe and happy old age without knowing calculus, for example. Children need to be exposed to a variety of math styles to find the one that suits them best.
But they also need to see meaningful to them people doing meaningful things with math and enjoying the experience. Math circles , where people help one another, are growing fast and are one way to achieve this. In his teaching, the superior man guides his students but does not pull them along; he urges them to go forward and does not suppress them; he opens the way, but does not take them to the place.
Guiding without pulling makes the process of learning gentle; urging without suppressing makes the process of learning easy; and opening the way without leading the students to the place makes them think for themselves. Now, if the process of learning is made gentle and easy and the students are encouraged to think for themselves, we may call the man a good teacher.
Only through education does one come to be dissatisfied with his own knowledge, and only through teaching others does one come to realize the uncomfortable inadequacy of his knowledge.
Being dissatisfied with his own knowledge, one then realizes that the trouble lies with himself, and realizing the uncomfortable inadequacy of his knowledge, one then feels stimulated to improve himself.
In contrast, the main intellectual resource for teachers in the United States is new ideas about education generated by educational research. Most U. Also, regular exchanges among teachers, which are common in China, are less frequent in U. Ma described two kinds of rewards that motivate U. One is salary, and the second is personal interest in students. The physical layout of Chinese classrooms differs from that of U. Chinese teachers have much larger classes: typically around twice the size of U.
When Ma was an elementary school teacher in China, she said, she had classrooms with as many as 60 students, and she has attended classes in China that have more than that. Also, classes in China typically have all of the desks facing the teacher, whereas in the United States, desks may be clustered into groups so that students can work together though many U.
When we teach math, we all focus on math. But in U. Also, in Chinese schools, students tend to stay in one room and teachers travel to that room to teach. But in China, students own their rooms and teachers travel to them. These rooms are primarily for work rather than for relaxation. They have desks, tables, a telephone, and bookshelves Figure Teachers use these rooms to grade homework, prepare and analyze les-.
Chinese teachers have fewer classes than do U. Outside the classroom, Chinese teachers spend considerable time grading homework. With 40 to 50 students per class, a teacher might have 90 homework assignments to correct each day. When a student is. Chinese teachers may have an hour and a half for lunch, compared with 20 to 30 minutes for many teachers in the United States.
In China, teachers often sit in on lessons given by other teachers and provide the teachers with comments after the class. Master teachers also give demonstration lessons to all of the mathematics teachers in a school. Every week there is time for mathematics teachers to discuss teaching, either in a whole group or divided by grades. Common topics for discussion are reflections about lessons, getting help with trouble spots, making connections between content areas, getting across difficult points, analyzing student errors and solution methods, and sharing successful experiences.
Sometimes, teachers also discuss mathematics education with mathematics teachers from other schools, education experts in universities, or master teachers from other cities. All of the mathematics teachers in the city of Suzhou, for example, have a specific time set aside on Thursday afternoons when they are not teaching so that they can meet to discuss mathematics education and see if they are progressing at the same pace.
Hongyan Zhao, a master teacher in Beijing, estimated that mathematics teachers in China spend about one-third of their time while in school on lesson planning and preparation, one-third of their time teaching, and one-third of their time engaged in discussions with other teachers including grading. Testing in China is focused on the tests given at the end of middle school and high school that dictate entrance into high schools and colleges, respectively.
In addition, at the local district or city level there is a universal final exam for mathematics given at the end of each academic year that is voluntary, though most schools participate. Testing, which has been used for thousands of years in China for government hiring, is viewed as less critical in China than in other countries such as the United States.
The prevailing view is that if students work hard they all have the potential to do well on tests. An important factor in judging teachers has been the success of their students on high school and college entrance exams, but more recently and in the larger cities, other aspects of teacher performance are being assessed. For example, peer evaluations and student evaluations are becoming more important. Mathematics education in China has been undergoing important changes in recent years.
Reform efforts patterned in part on the standards. Hongyan Zhao, a master teacher at the upper secondary school attached to Tsinghua University in Beijing, described a busy but not unusual day in her life as a teacher. She arrived at the school at a.
First, she went over the lesson plan before going to class, reviewing the best way to teach her students about linear equations. Chinese students begin learning their maths facts at a very early age: maths textbooks begin with multiplication in the first semester of second grade, when children are seven years old.
Stemming from this tradition, most classrooms have few concrete teaching materials for maths lessons. The cultural traditions of Chinese maths education lead people to believe that routine practice is the most efficient way to learn.
This continues today. And as a result, schools in Shanghai have scored highly in recent years on international tests of maths ability. It is this aptitude for maths among Chinese schoolchildren that has led the UK government to announce plans to bring over 60 maths teachers from Shanghai to help teach in centres of excellence.
The Chinese curriculum in maths is a nine-year programme divided into four mathematical stages, running from primary school to grade 9, when a child is 14 years old. The curriculum sets out four teaching periods a week for maths in primary and junior high schools.
However, most schools arrange more than five periods each week. A typical teaching period in primary schools is approximately 40 minutes, extending to 45 minutes in secondary school. Teachers often set at least half an hour of homework every day for primary school pupils and more for secondary pupils.
The goal of maths education in China is to develop conceptual and procedural knowledge through rigid practice.
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