THE TEACHING OF MATHEMATICS

Math is definite, logical and objective. The rules for determining the truth or falsity of a statement are accepted by all. If there are disagreements, it can readily be tested. It is in contrast with the subjective characteristics of other subjects like literature, social studies and the arts.

Math deals with solving problems. Such problems are similar to all other problems anyone is confronted with. It consists of: a) defining the problem, b) entertaining a tentative guess as the solution c) testing the guess, and d) arriving at a solution.

Nature of Mathematics

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The strategy for teaching Mathematics depends on the objectives or goals of the learning process. In general these goals are classified into three: a) knowledge and skill goals, b) understanding goals and c) problem solving goals.

Strategies in Teaching

Mathematics

Knowledge and Skill Goals

Knowledge and basic skills compose a large part of learning in Mathematics. Students may be required to memorize facts or to become proficient in using algorithms.

Ex. of facts: 2 X 10= 20 Area of rectangle = B x H

Ex. of skills: Multiplying two-digit whole numbers

Changing a number to scientific notation

Knowledge and skill goals require automatic responses which could be achieved through repetition and practice.

Strategy Based on Objectives

Understanding GoalsThe distinguishing characteristics of

understanding goal is that “understanding must be applied, derived or used to deduce a consequence”. Some strategies used in understanding are:

a. authority teachingb. interaction and discussionc. discoveryd. laboratorye. teacher-controlled presentations

a.) Authority teachingThe teacher as an authority

simply states the concept to be learned. The techniques used are by telling which is defined, stating an understanding without justification, by analogy, and by demonstration.

b.) Interaction and discussionInteraction is created by asking

questions in order to provide means for active instead of passive participation.

c) Discovery The elements of a discovery

experience are motivation, a primitive process, an environment for discovery, an opportunity to make conjectures and a provision for applying the generalization.

d) LaboratoryThe advantages are: a) maximizes student participation, b) provides appropriate level of difficultyc) offers novel approachesd) improves attitudes towards

mathematicsThis is done through experimental

activities dealing with concrete situations such as drawing, weighing, averaging and estimating. Recording, analyzing and checking data enable students to develop new concepts and understanding effectively.

e) Teacher-controlled presentationsThe teacher uses educational

technology such as films and filmstrips, programmed materials, and audio materials. Other activities are listening to resource persons and conducting field trips. Suitable places for educational trips are government agencies such as the weather bureau, post office and community supermarkets, factories and transportation centers like the bus depot and airport.

Problem-solving Goals

Problem solving is regarded by mathematics educators and specialists as the basic mathematical activity. Other mathematical activities such as generalization, abstraction, and concept building are based on problem solving. Others believe that the more important roleof problem solving in the school curriculum is to motivate all students not only those who have a special interest in mathematics and a special aptitude for it.

1. Problem Solving

2. Concept Attainment Strategy

3. Concept Formation Strategy

Strategies in Teaching Mathematics

1. Problem solving

Theoretical Basis for Problem-solving Strategy Constructivism – This is based on Brunner’s

theoretical framework that learning is an active process in which learners construct new ideas or concepts based upon current/past knowledge.

Cognitive theory – The cognitive theory encourages students’ creativity with the implementation of technology such as computer which are used to create practice situations.

Guided Discovery Learning

Tool engages students in a series of higher order thinking skills to solve problems.

Metacognition TheoryThe field of metacognition process holds that

students should develop and explore the problem, extend solutions, process and develop self-reflection. Problem solving must challenge students to think.

Cooperative learning

The purpose of cooperative learning group is to make each member a stronger individual in his/her own right. Individual accountability is the key to ensuring that all group members are strengthened by learning cooperatively. Teachers need to assess how much work each member is contributing to the group’s work, provide feedback to groups and individual students, help groups avoid redundant efforts by members, and make sure that every member is responsible for the final outcome.

The favorable outcomes in the use of cooperative learning is that students are taught cooperative skills such as: a) forming groups, b) working as a group, c) problem solving as a group and d) managing differences

Steps of the Problem Solving Strategy

1. Restate the problem 2. Select appropriate notation. It can help them

recognize a solution. 3. Prepare a drawing, figure or graph. These can help

understand and visualize the problem.4. Identify the wanted, given and needed information.5. Determine the operation to be used.6. Estimate the answer.

Knowing what the student should get as the answer to the problem will lead the students to the correct operations to use and the proper solutions.

7. Solve the problem. The student is now ready to work on the problem.8. Check the solution. Find a way to verify the solutions in order to experience the process of actually solving the problem.

Other Techniques in Problem Solving1. Obtain the answer by trial and error.

It requires the student to make a series of calculations. In each calculation, an estimate of some unknown quantity is used to compute the value of a known quantity.

2. Use an aid, model or sketch.A problem could be understood by

drawing a sketch, folding a piece of paper, cutting a piece of string, or making use of some simple aid. Using an aid could make the situation real to them.

3. Search for a patternThis strategy requires the students to

examine sequences of numbers or geometric objects in search of some rule that will allow them to extend the sequences indefinitely.

Example: Find the 10th term in a sequence that begins, 1, 2, 3, 5, 8, 13, . . . . .

This approach is an aspect of inductive thinking-figuring a rule

from examples.

4. Elimination Strategy

This strategy requires the student to use logic to reduce the potential list of answers to a minimum. Through logic, they throw away some potential estimates as unreasonable and focus on the reasonable estimates

Concept attainment strategyThis strategy allows the students to

discover the essential attributes of a concept. It can enhance the students’ skills in (a) separating important from unimportant information; (b) searching for patterns and making generalizations; and (c) defining and explaining concepts.

Steps a. Select a concept and identify its essential

attributes b. Present examples and non-examples of

the concept c. Let students identify or define the

concept based on its essential attributes d. Ask students to generate additional

examples

(Sample Activity on Fractions)

Effective use of the concept attainment Strategy

The use of the concept attainment strategy is successful when:

a. students are able to identify the essential attributes of the concept

b. students are able to generate their own examples

c. students are able to describe the process they used to find the essential attributes of the concept

Concept Formation Strategy

This strategy is used when you want the students to make connections between and among essential elements of the concept:

Stepsa. Present a particular question or problem.

b. Ask students to generate data relevant to the question or problem.

c. Allow students to group data with similar attributes.

d. Ask students to label each group of data with similar attributes.

e. Have students explore the relationships between and among the groups. They may group the data in various ways and some groups maybe subsumed in other groups based on their attributes.