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Nature 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.
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.… Strategies in Teaching Mathematics The strategy for teaching Mathematicsdepends on the objectives or goals of thelearning process. In general these goals areclassified into three: a) knowledge and skillgoals, b) understanding goals and c) problemsolving goals.
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Strategy Based on ObjectivesKnowledge 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 HEx. 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.
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Understanding Goals The 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 teaching b. interaction and discussion c. discovery d. laboratory e. teacher-controlled presentations
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a.) Authority teaching The teacher as an authority simplystates the concept to be learned. The techniquesused are by telling which is defined, stating anunderstanding without justification, byanalogy, and by demonstration.b.) Interaction and discussion Interaction is created by askingquestions in order to provide means for activeinstead of passive participation.
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c) Discovery The elements of a discovery experienceare motivation, a primitive process, anenvironment for discovery, an opportunity to makeconjectures and a provision for applying thegeneralization.
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d) Laboratory The advantages are: a) maximizes student participation, b)provides appropriate level ofdifficulty c) offers novel approaches d) improves attitudes towards mathematics This is done through experimentalactivities dealing with concrete situations such asdrawing, weighing, averaging and estimating.Recording, analyzing and checking data enablestudents to develop new concepts andunderstanding effectively.
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e) Teacher-controlled presentations The teacher uses educational technologysuch as films and filmstrips, programmedmaterials, and audio materials. Other activities arelistening to resource persons and conducting fieldtrips. Suitable places for educational trips aregovernment agencies such as the weatherbureau, post office and communitysupermarkets, factories and transportation centerslike the bus depot and airport.
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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.
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Strategies in Teaching Mathematics1. Problem Solving2. Concept Attainment Strategy3. Concept Formation Strategy
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1. Problem solvingTheoretical 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.
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Guided Discovery Learning Tool engages students in a series of higher order thinking skills to solve problems. Metacognition Theory The 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.
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Cooperative learning The purpose of cooperative learning group is to makeeach member a stronger individual in his/her own right.Individual accountability is the key to ensuring that all groupmembers are strengthened by learning cooperatively.Teachers need to assess how much work each member iscontributing to the group’s work, provide feedback to groupsand individual students, help groups avoid redundant effortsby members, and make sure that every member isresponsible for the final outcome. The favorable outcomes in the use of cooperativelearning is that students are taught cooperative skills suchas: a) forming groups, b) working as a group, c) problemsolving as a group and d) managing differences
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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.
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7. Solve the problem. The student is now ready to work on the problem.8. Check the solution. Find a way to verify the solutionsin order to experience the process of actually solving theproblem.
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Other Techniques in Problem Solving 1. 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.
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3. Search for a pattern This 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.
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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
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Concept attainment strategy This 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.
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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
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Effective use of the concept attainment StrategyThe 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
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Concept Formation Strategy This strategy is used when you want the students to make connections between and among essential elements of the concept:
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Steps a. 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.