Ebbutt and Straker (1995) mention the characteristics of school mathematics as follows:
1. Mathematics as search activity patterns of relationships.
The implications of this view of learning are: (a) giving students the opportunity to conduct discovery and investigation to determine patterns of relationships, (b) providing opportunities for students to experiment with a variety of ways, (c) encouraging students to discover the existence of sequence, difference, comparison, grouping, etc., (d) encouraging students to draw general conclusions, and (e) helping students understand and find the relationships between understanding one another.
The implications of this view of learning are: (a) giving students the opportunity to conduct discovery and investigation to determine patterns of relationships, (b) providing opportunities for students to experiment with a variety of ways, (c) encouraging students to discover the existence of sequence, difference, comparison, grouping, etc., (d) encouraging students to draw general conclusions, and (e) helping students understand and find the relationships between understanding one another.
2. Mathematics as requiring imagination, creativity, intuition and invention.
The implications of this view of learning are: (a) encouraging the initiative and providing an opportunity to think differently, (b) encouraging curiosity, the desire to ask, deny the ability, and the ability estimation, (c) appreciating the unexpected discovery as it is useful to think of it as an error, (d) encouraging students to discover the structure and design of mathematics, (e) encouraging students to appreciate the discovery of other students, (f) encouraging students to think reflexively, and ( g) not recommending just one method only.
The implications of this view of learning are: (a) encouraging the initiative and providing an opportunity to think differently, (b) encouraging curiosity, the desire to ask, deny the ability, and the ability estimation, (c) appreciating the unexpected discovery as it is useful to think of it as an error, (d) encouraging students to discover the structure and design of mathematics, (e) encouraging students to appreciate the discovery of other students, (f) encouraging students to think reflexively, and ( g) not recommending just one method only.
3. Mathematics as problem solving activities (problem solving)
The implications of this view of learning are: (1) providing an environment that stimulates math learning problems, (2) helping students solve math problems using his own way, (3) helping students learn the necessary information to solve math problems, (4) encouraging students to think logically, consistently, systematically, and to develop documentation systems/records, (6) helping students learn how and when to use various teaching aids/mathematical educational media, such as: terms, calculators, etc..
The implications of this view of learning are: (1) providing an environment that stimulates math learning problems, (2) helping students solve math problems using his own way, (3) helping students learn the necessary information to solve math problems, (4) encouraging students to think logically, consistently, systematically, and to develop documentation systems/records, (6) helping students learn how and when to use various teaching aids/mathematical educational media, such as: terms, calculators, etc..
4. Mathematics as a means of communication
The implications of this view of learning are: (1) encouraging students to recognize the nature of mathematics, (2) encouraging students to make examples of the nature of mathematics, (3) encouraging students to explain the nature of mathematics, (4) encouraging students to justify the need for math activities, (5) encouraging students to discuss math problems, (6) encouraging students to read and write mathematics, (7) respecting for students' native language in discussing mathematics.
Meanwhile, according to Bell (1998), the direct object of the lesson of school mathematics can be classified as follows:
a. Fact
Fact is the agreement or convention made in mathematics, for example terms (names), the notation (symbols), and agreement (convention).
Example:
- Notation or symbol “ ^" for the word "and" in mathematical logic.
- An agreement "On the number line, the right of “0” is positive, and the left of “0” is negative.
How to teach the facts can be done with various techniques, among others are memorization, drill, demonstrations, etc.
b. Concept
Concept is an abstract notion or idea that allows someone to classify the object or event and to determine whether an object or event is an example of abstract ideas or not.
Example:
- The concept of function is described with or without examples.
- The concept of the natural numbers: 1, 2, 3, 4, ....
Several concepts are fundamental understanding that can be captured naturally, vividly, and without having to be defined. Example: set, dots, etc..
Meanwhile, another concept is explained, defined or given a constraint using the previous concepts. For example, the understanding of prime numbers is explained using the understanding of factors (prime numbers are numbers which have exactly two factors). Understanding the factors is described as part of the multiplication. Understanding multiplication is repeatedly described as a summation. Understanding the sum is described as a merger of two disjoint sets. So, the concepts that form a network concept are also called concept maps.
How to teach the concept:
Start with the belief that the students already have the prerequisite knowledge and deductive approaches, as well as inductive, or it could be the perception, abstraction and generalization.
c. Principle
Principle is a statement which states the entry into force of a relationship between some of the concepts. The statement may declare the properties of a concept, laws, theorems, or propositions true in that concept. Similar to concepts, principles are also tiered.
Example:
- The sum of the first –n natural numbers is (½)n (n + 1)
- Rectangle can occupy exactly the frame with 4 ways.
How to teach principles:
Identify the concepts which are already known, then use a process of inquiry, guided discovery, group discussions, problem solving, demonstrations, etc.
d. Operation/procedure
Operation or procedure is the working steps in mathematics, for example, the steps in multiplying compound, procedure, and solving the equation. This procedure is also called algorithms. This procedure is to accelerate progress, but still based on the correct logic. Therefore, this object is also called as a skill.
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