This paper selects four mathematical problems: abdication subtraction, multi-digit multiplication, fractional division, and the relationship between the area and perimeter of closed graphs. Every problem is a research topic. First, design the classroom situation, and then ask American and China teachers to explain the teaching methods and solutions about this content. Amazingly, Dr. Ma can deeply analyze these accustomed methods and reflect the mathematical principles behind them. Finally, through comparison, talk about the knowledge differences between teachers in the two countries.
Feeling: Two words that puzzled me: "process understanding and concept understanding". The difference between borrowing 1 and taking 1 is that taking 1 is a process, while taking 1 is a conceptual principle. What I gain more is a better understanding of "knowledge network" and "knowledge package" after reading. For example, on page 1 14 of the book, there are three contrast pictures, which are four hand-in-hand pictures drawn from the original completely separated knowledge points, with implicit basic attitude towards mathematics and explicit practice of basic principles. It reflects his profound understanding of mathematics, including width, depth, relevance and integrity. Besides schema explanation, there are more vivid metaphors. In order to master knowledge, we must know the city like a skilled taxi driver. There are many roads and a complicated map in our mind.
Chapter 1: the difference between recombination number and subtraction;
For example, American teachers think that 7+2+3=9= 12 is right, and even China students will not make such mistakes.
American teacher's method: "Take 1 10 and replace it with 10." Take is reorganization, change is transformation. ""You can't subtract a larger number from a smaller number, so you borrowed it from its neighbor 1. "There are many mathematical mistakes in this sentence. Decimals can also be approximated by large numbers. Neighbors suggest that ten digits and four digits are two independent numbers, not two parts of a number. " Borrowing number implies that the numerical value in calculation can be changed at will, and the numerical value can be borrowed.
China's teacher's method: "retreat one" corresponds to "advance one". Use "decimal" and "bit value" to explain. Various reorganization methods are also introduced. Such as 53-26:53 divided into 40 and13; 53 divided by 40, 10 and 3 or 26 divided by 20, 3 and 3, etc. Special emphasis is placed on knowledge packages. When doing 53-26, they already have the foundation of "subtraction within 20" and must learn it very solidly. They say that the reorganization idea of retreating the high position to the low position in subtraction is formed by learning three levels of problems. Within 20 and 100, the greater the minuend. When teaching a knowledge point, we should regard knowledge as a package and know the role of current knowledge in the knowledge package. You also need to know what concepts or processes support the knowledge you teach, so your teaching depends on, strengthens and describes the learning of these concepts in detail.
Contrast: American teachers prefer "borrowing". Teacher China explained that this algorithm is "returning one".
Chapter 2: Multiplication: Dealing with Students' Errors
Wrong question: the cubic product of 123 is aligned with the unit.
×645
6 15
492
738
1845
American teachers believe that the reason for the mistake is that students don't understand the bit value well, and on the surface it is only regarded as 4 times 3. Will not transfer, did not remember the accepted rules. What American teachers don't know is that product 492 is actually 4920. Think that zero is "artificial". Just for space.
Strategy: procedural:
Description rule: 5 in the unit starts with the unit; The tenth digit 4 moves to the left, and the product is placed below the tenth digit 4. ; Then move to a hundred places. And so on.
Use paper with horizontal lines: stand the paper upright and write a number in a column. So as to ensure that these intermediate products can be arranged correctly.
Use placeholders: put an apple, an orange and an elephant in the blank. Just obey the teacher's interesting and arbitrary orders.
Explain the basic principle: 123 is 100 plus 20 plus 3,5123,40123,600123.
The problem is divided into three small problems: 123 multiplied by 5, 40 and 600 respectively, and the products are 6 15, 4920 and 73800, and then added.
China teachers think:
Interpretation error: change the distribution law:123× 645 =123× (600+40+5) to the detailed vertical type, and then erase the stepped vertical type.
10 and 10, × 10 and×100 are multiplied by the power.
Value system: prove that 123 times 4 10 equals 492 10, and explain why 492 should be aligned with 10. The size of a number depends not only on the number it contains, but also on the number of digits placed in these numbers.
Knowledge package: bit value, meaning of multiplication, basic principle of multiplication, multiplication of two digits, multiplication of one digit, multiplication of 10, multiples and powers of 10, commutative law. The emphasis is on the multiplication of two digits. (omitted in fig. P44)
Strategy: Explain: Where should 492 tens and twos be written? Where should I write 738 108?
Students find problems:
Observation, inspection, analysis and discussion; Guide by asking questions; Diagnostic exercises.
Contrast: American teachers follow the principle of "alignment multiplier", while China uses the concepts of bit value and bit system to explain why intermediate products are not aligned as addition.
Chapter four: Exploring new knowledge: the relationship between perimeter and area.
When the perimeter of a closed figure increases, its area also increases. Judgment and proof.
American teacher: look up books. Because he doesn't understand the operational derivation of the formula, he needs someone to tell him counterexamples, consult books and find counterexamples. More examples and mathematical methods are needed. Response to students' views: simply accept 9%, learn 78% without mathematical methods, and learn with mathematical methods 13%. Teachers evaluate the correctness of students' views responsibly and study the correctness of their own views with students.
Miss China: 70% can solve it correctly. Overturning this proposition: the first level of understanding. Identifying possibility: the second level of understanding. Clarification conditions: the third level of understanding. Explanatory conditions: the fourth level of understanding.
Contrast: American teachers tend to pay attention to the view that "the perimeter increases and the area increases". Teacher China discusses the relationship between perimeter and area. Three American teachers (13%) did the research independently, and only one got the correct answer. In China, 66 (92%) teachers did research on mathematics, and 44 (62%) teachers got the correct answers. Mastering the basic ideas of a field includes not only mastering the general principles, but also developing the possibility of learning and investigation, guessing and intuition, and the attitude of solving problems independently.
Only those teachers who adapt to mathematics can cultivate students' ability to explore mathematics. Teachers in China are more culturally adapted to this subject. They tend to think critically, discuss in mathematical terms, and judge their views with mathematical arguments.