Fractional percentage

To understand the meaning of decimals, we can start with the meaning of fractions, which can be explained by subdivision and comprehensive activities. When a whole (reference quantity) is divided into equal parts, the quantity in which some parts gather is called "component", and this "component" is expressed or recorded by "fraction". For example, 2/5 refers to the "component" in which an integer is divided into five equal parts and then aggregated into two halves. When the whole is divided into 10, 100, 1000, etc. Use another method to record the weight at this time-decimal. For example,110 is 0. 1, 2/ 100 is 0.02, 5/ 1000 is 0.005 and so on. The "."is called the decimal point, which is used to separate the integer part from the decimal part that cannot form an integer. Integers that are not 0 are called decimals, and those that are 0 are called pure decimals. So the meaning of decimals is part of the meaning of fractions.

Second, the decimal structure

Decimal numeration system describes its rules by associating written symbols with physical quantities. Before the decimal point (left) is used to indicate the number of integer parts. The first integer is the quantity that records how many 1 are in the integer, and this position is called a unit; The second integer record counting from the decimal point is the amount of ten, which is called ten digits; ..... and so on. The decimal point is counted backwards (to the right), indicating the number of decimal parts (less than 1). The first decimal place is how many decimals there are in the recording element, and this position is called decimals; The second digit after the decimal point is how many percentiles there are in the record, which is called percentile, and so on. In the multi-unit counting system of numbers, "ten", "unit", "tenth" and "percentile" are called "place names"; The numerical values it represents, such as ten, one, 0. 1 and 0.0 1, are called "bit values". "Ten", "one", "0. 1" and "0.0 1" can be used as numeration units.

In addition, "number" can also be expressed in different numeration units, such as "one", "0. 1" and "0.0 1". Judging from the above decimal structure, it is very important for students to construct the decimal structure and the concept of decimal value.

Third, the cognitive process of decimal learning

(a) hibbert and Wayne's "Theory of Mathematical Symbol Writing Ability Development"

1. Link process

You can use indicators familiar to students to create links with mathematical symbols. For example, decimal symbols can be derived from articles in life (such as currency, metric measurement, etc.). ) or teaching AIDS (such as math building blocks), so that when children see "1.8" in the future, they will have "1 glass of water, 0.8 glass of water" in their hearts.

2. Color development method

The development process refers to the process that students use the manipulation of indicators to deal with the developed symbols. For example, through the manipulation of building blocks, students learn that decimal notation will be generated when units are represented by "bars", and then they find that the amount less than one unit is represented by fractions as well as decimals.

3. Refining process

Refinement is the process of extending grammar programs to other appropriate situations. For example, students learn from the building blocks that when the unit is "bar", there will be a decimal place. The refining process can be further classified to two decimal places.

4. Conventional process

If children often practice grammar programs, they can use mathematical symbols to solve problems more effectively.

5. Construction process

Students take the mathematical symbols and rules they have learned before as the indicators of the new mathematical symbol system, and then cycle the above four cognitive processes again to establish a more abstract mathematical symbol system.

D'Entremont's "Onion Model of Decimal Learning"

Doernte Monte believes that the cognitive process of decimal learning includes five different levels, and each level is gradually surrounded by external levels. Conceptual knowledge is the core of decimal knowledge. In order to acquire the conceptual knowledge of decimals, students must peel off the upper epidermis layer by layer.

1. Hierarchy of concrete objects

The first level children encounter is the level of concrete objects. Teachers guide students into the decimal world through visible objects in the real world. For example, we can use building blocks to introduce the concept of decimal places. If we regard a building block as the unit "1", then a building block is regarded as "0. 1".

2. Level of operating instructions

The teacher changed from teaching with concrete objects to teaching with decimal notation. The teaching content includes the introduction of decimal notation and how to apply decimal notation.

3. The level of the program

Students can not only use symbols alone to calculate decimals, but also follow the calculation rules of decimals. But I don't reflect on what steps I just took. Therefore, even if a student calculates, it does not mean that the student must understand the meaning behind it.

4. The level of mental model

At the level of mental model, students will not blindly follow formulas, and they can clearly know the reasons when solving problems.

5. The level of abstraction

At this time, students have a good intuition about decimals, and they no longer need visible objects to help them understand. They can combine "how to deal with decimals" with "why". At this stage, children can understand the concept of decimal, which is the core of decimal knowledge.