We all have a thing that we can’t help doing almost every day. This is nothing but mathematics. We would like to listen to music; this is just mathematics because the rhythmic beats of melody make us pleasant in our hearing, and how many beats offer pleasure to us relies on mathematics, as Dr Khin Maung Win, who is U Khin Maung Latt and Daw Khin Myo Chit’s son, said. Of course, we must number items or calculate costs nearly daily. These are mathematics only. So, even though we do not like mathematics at all, this subject is impossible to be negligible in one way or another. We can study essential mathematics based on Gagne’s Information Processing Model of Learning. Now, this model will be explained in the following.

In his book Essentials of Learning for Instruction (1975), Gagne forwarded the theory concerning how humans obtain information during a learning process. According to his learning theory, stimuli from the external environment will activate the nervous system through the human sensory organs. This information will be interpreted in the short-term memory, encoded transit and stored in the long-term memory in conceptual form. When retrieved, it will first enter the response operator, which can decide, control and implement the form of human behaviour that interacts with the environment. According to Gagne, experiences kept in long-term memory are important for humans to facilitate the process of new learning. Founded on his theory, Gagne later ascertained and suggested the eight phases usually experienced by the human mind during a certain learning process. Those eight phases are motivation, comprehension, storing, retention, recall, generalization, performance, and feedback.

Firstly, signal learning is the most primitive form of learning, and it can be grouped under the classical conditioning theory of learning, as explained by Pavlov. For example, humans learn the signal of smoke as fire and that of lightning as thunder through experience. Secondly, stimulus-response learning is related to stimuli that have been planned and stimulated with individuals’ responses in order to produce changes in their behaviour, which can be classified under the operant conditioning suggested by Skinner. For instance, the teacher shows a pyramid object for students to read out `pyramid’. Thirdly, chain learning refers to various relationships which occur after the process of stimulus-response learning with the aim of upgrading the learning stage to a higher level. For example, one is able to make sentences by relating words such as `Health is a gift of nature’. Fourthly, verbally associated learning is the primary form of learning a language. One example of this learning is the name of a person with his address, such as `Saw Thae Aung from Thebwet’. Fifthly, multiple discrimination learning means selecting one response only from various stimuli in learning. One instance of this learning is that after learning the difference in pronouncing the scientific name Amoeba in Biology and the person’s name Moe Moe in Myanmar, it will enable a student to pick one only to pronounce `moe’ in Amoeba. Sixthly, conceptual learning is referred to as learning in forming concepts with representative symbols based on the same characteristics. For example, birds, chickens, and ducks belong to the bird family, and cats, tigers, and lions are also related to the cat family. Seventhly, principal learning is the relationship between two or more concepts that have been learned. For instance, `Water flows to a lower level’ is a principle in which water is seen to be able to flow and take space in any lower place. Eighthly, problem-solving learning is learning through the thinking process by using concepts and principles which have been learnt. For example, the exact height of a pyramid in Egypt can be calculated in accordance with similar triangles, namely by Thales.

According to Gagne, the four important categories which must be mastered by students in mathematics are fact, skill, concept and principle. Mathematical facts are the language of mathematics, such as symbols to present numbers (e.g. 0, 1, 2, 3), operation signs (e.g. +, -, ×, ÷), and Greek alphabets (e.g. alpha, beta, theta, phi). These mathematical facts can learned through memorization, verbal or written practice, and games under stimulus-response learning. Mathematical skills are procedure operation-out accuracy in a reasonable, appropriate time. Examples of such skills include constructing the right angle and drawing circles, which are mastered through practice and games. As such, this type of learning is actually chain learning. Mathematical concepts are abstract ideas derived from concrete examples, such as definitions given in the form of set and perimeter. Those concepts can be done through understanding definitions or observations of concept-related objects. Hence, this learning is called conceptual learning. Finally, the mathematical principle is an integration of and relationship between the concepts of mathematics. These concepts can be learned through the process of inquiry-discovery or problem-solving. A student is said to have mastered a certain principle when he can ascertain the concepts contained in the principle, relate these concepts according to a suitable sequence, and apply this principle in some specific situation. This kind of learning is simply problem-solving, which is considered the most complex level of learning.

Bruner’s other theorems of learning mathematics are still left that should be studied, as follows.

Theorem of construction: The most effective way for a student to master a certain mathematical concept, principle or law is to construct a representation to express this mathematical concept, principle or law, where practical activities should be carried out. For example, a student might master the commutative law using the discovery method to obtain the law by means of such a few operations as addition and multiplication.

Theorem of notation: Mathematical notations should be introduced according to a student’s cognitive development. For example, in the teaching of algebraic equations, it is better to start with notations such as y = 2x + 3. After mastering this, the equation y = f(x) could be introduced at the upper secondary school level.

Theorem of contrast and variation: The procedure to introduce abstract representation from concrete representation involves contrast and variation operations. Most mathematical concepts will not be meaningful if they cannot be distinguished from other concepts. For example, concepts like curve, radius, and diameter can be more meaningful only if their characteristics can be distinguished. Other than this, every new mathematical concept should be introduced with various types of examples — 3x + 4x = ?, 3x + ? = 7x, ? + 4x = 7x.

The theorem of relation: Every mathematical concept, principle and skill ought to relate to other concepts, principles and skills. During the teaching process, a teacher should use existing concepts, principles or skills to form new ones. For example, the operation in multiplication goes well only after that in addition. And the operation of differentiation is inversely related to that of integration.

Some students fear mathematics like a ghost, while others get bored with doing mathematics. Then, they will be in poor mathematics quite undesirably. In fact, students should not do badly in mathematics in one way or the other. What I learned from an article in Mingala Maung Mel Magazine, as far as I can remember, is that someone has to familiarize themself with anything that he is afraid of. Really, we can learn mathematics by doing only. Students who are not interested in mathematics are also given the opportunity to solve mathematical problems with a will. In actual fact, when students learn mathematical sums, they need to understand mathematical concepts all first but not memorize them by heart. Second, they had better do these sums all by themselves. Lastly, they should frequently do their well-learnt mathematical calculations very repeatedly. If they wish, they can try to find solutions to unseen math problems. Mathematics is the language of physics, as the old curriculum on Grade 9 Physics stated. That is, mathematics and physics often go hand in hand, and it will not be difficult for a student to become an outstanding one in Physics if he has a good command of math. Clearly, mathematics takes an enormous role not only in physics but also in other fields of study, such as engineering, chemistry, research, medicine, meteorology, hydrology, and even literature. Finally, there is only one thing that I want to say once again. Learn mathematics by doing only!

# Learning Mathematics Oriented to Gagne’s Model

- October 15, 2024
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