The Object Of Mathematics Learning o411d

The Object of Mathematics Learning Before examining Gagne’s four phases of a learning sequence and eight types of learning, it is appropriate to discuss the objects of mathematics learning, which are considered in his theory. These objects of mathematics learning are those direct and indirect things which we want student to learn in mathematics. The direct objects of mathematics learning are facts, skills, concepts, and principles; some of the many indirect objects are transfer of learning, inquiry ability, problem-solving ability, self-discipline, and appreciation for the structure of mathematics. The direct objects of mathematics learning – facts, skills, concepts, and principles – are the four categories into which mathematical content can be separated. Mathematical facts are those arbitrary conventions in mathematics such as the symbol of mathematics. It is a fact that 2 is the symbol for the word two, that + is the symbol for the operation of addition, and that the sine is the name given to a special function in trigonometry. Facts are learned through various techniques of rote learning such as memorization, drill, practice, timed test, games, and contests. People are considered to have learning a fact when they can state the fact and make appropriate use it in a number of different situations. Mathematical skills are those operations and procedures which students and mathematicians are expected to carry out with speed and accuracy. Many skills can be specified by sets of rules and instructions or by ordered sequences of specific procedures called algorithms. Among the mathematical skills which most people are expected to master in school are long division, addition of fractions and multiplication of decimal fractions. Constructing right angles, bisecting angles, and finding unions or intersection of sets of objects and events are examples of other useful mathematical skills. Skills are learned through demonstrations and various types of drill and practice such as worksheets, work at the chalkboard, group activities and games. Students have mastered when they can correctly demonstrate the skill by solving different types problems requiring the skill or by applying the skill in various situations. A concept in mathematics is an abstract idea which enables people to classify objects or events and to specify whether the objects and events are examples or non-examples of the abstract idea. Sets, subsets, equality, inequality, triangle, cube, radius and exponent are all examples of concepts. A person who has learned the concept of triangle is able to classify sets of figures into subsets of triangles and non-triangles. Concepts can be learned either through definitions or by direct observation. By direct observation and experimentation young children

learn to classify plane objects into sets of triangles, circles, or squares; however few young children would be able to define the concept of a triangle. A concept is learned by hearing, seeing, handling, discussing, or thinking about a variety of examples and non-examples of the concept and by contrasting the examples and non-examples. Younger children who are in Piaget’s stage of concrete operations usually need to see or handle physical representations of a concept to learn it; whereas older formal operational people may be able to learn concepts through discussion and contemplation. A person has learned a concept when he or she is able to separate example of the concept from non-examples. Principles are most complex of the mathematical objects. Principles are sequences of concepts together with relationships among these concepts. The statements, “triangles are congruent if two sides and the included angle of one triangle are equal two sides and the included angle of the other” and “the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides” are examples of principles. Each of these principles involves several concepts and relationships among these concepts. To understand the principle about congruent triangles, one must know the concepts triangle, angle, side. According to Gagne (1996) in a chapter appearing in the book Analyses of Concept Learning edited by Herbert J. Klausmeier and Chester W. Harris: It would appear, then, that principles can be distinguished from what have previously been called concepts in two ways. First, the performance required to demonstrate that a concept has been learned is simply an identification, that is a choice from a number of alternatives; a principle, in contrast, must be demonstrated by means of performances that identify its component concepts and the operation relating them to one another. Second, this means that the inference to be made about mediating processes is different in the two cases. A concept is a single mediator that represents a class of stimuli (or objects), whereas a principle is a sequence of mediators, each one of which itself a concept. (pp. 86-87) Principles can be learned through processes of scientific inquiry, guided discovery lessons, group discussions, the use of problem solving strategies and demonstrations. A student has learned a principles when he or she can identify the concepts included in the principle, put the concepts in their correct relation to one another, and apply the principle to a particular situation. It probably would not be a very precise or useful activity to classify all the objects of secondary school mathematics into the four object categories – fact, skills, concepts, and

principles. Even the experts in mathematics and learning theory would disagree about the proper category for many mathematical objects. In general, the objects progress in order complexity from simple facts, to skills an concepts, through complex principles. Also the classification of many (maybe even most) mathematical objects is relative to the observer’s viewpoint, which is an important fact (or is that a principle!) for every mathematics teacher to know. A student who merely memorizes the quadratic formula and come up with two answers has learned a skill. A

student who can classify 5, 3, and 4 as constants and

equation

as a variable for the quadratic

is demonstrating acquisition of a concept. And, a person who can

derive (or prove) the quadratic formula and explain his derivation to someone else has mastered a principle. Consequently, the quadratic formula which is a principle may be regarded as either a fact, a skill, or a concept by a student whose viewpoint of the quadratic formula is not as sophisticated as that of a mathematician. As a mathematics teacher, you should develop testing and observation techniques to assist you in recognizing students’ viewpoints of the concepts and principles which you are teaching. All of us have at times memorized the proofs of theorems, with no understanding of the concepts and principles involved in the proof, in order to tests. While this subterfuge is a form of learning, it is not what teachers hope to have students learn by proving theorems. The point to recognize here is that many times when teachers are teaching what they view as mathematical principles, student are internalizing as facts or skills the information which being presented.