Methods of teaching Mathematics

Methods of Teaching MATHEMATICS contents next Genalyn R. Obias Marian Angeli A. Palma

LAGUNA STATE POLYTECHNIC UNIVERSITY Siniloan (Host) Campus VISION A premier university at CALABARZON offering academic programs and related services designed to respond to the requirements of the Philippines Economy particularly Asian countries. next back contents

MISSION The university shall primarily provide advance educational professions, technical and vocational instructions in agriculture, fisheries, forestry, science engineering, industrial technology, teacher’s education, medicine, law, arts and sciences, information technology, and other related fields. It shall undertake research and extension services provide progressive leadership in its area of specialization. next back contents

GOALS In pursuit of the college vision/mission the college of education is committed to develop potentials the full potentials of the individuals and equip them with knowledge, skills and attitudes in Teacher Education allied fields to effectively respond to the increasing demands, challenges and opportunities of changing time for global competitiveness. next back contents

OBJECTIVES Produce graduates who can demonstrate and practice the professional and ethical requirement for the Bachelor of Secondary Education such as; 1. to serve as positive and role model in the pursuit of learning thereby maintaining high regard to professional growth; 2. focus on the significance of the providing wholesome and desirable learning environment; next back contents

3. facilitate learning process in diverse type of learners; 4. use varied approaches and activities, instructional materials, and learning resources; 5. use assessment date to plan and revise the teaching learning plans; 6. direct and strengthen the links between schools and community activities; 7. conduct research and development in teacher education and other related activities. back next contents

FOREWORD This Teacher’s Module entitled “ Methods of Teaching Mathematics ” is part of the requirements in Educational Technology 2 under the revised Education curriculum based on CHED Memorandum Order (CMO)-30, Series of 2004. Educational Technology 2 is a three (3)-unit course designed to introduce both traditional and innovative technologies to facilitate and foster meaningful and effective learning where students are expected to demonstrate a sound understanding of the nature, application and production of the various types of educational technologies. contents back next

The students are provided with guidance and assistance of selected faculty members of the College through the selection, production and utilization of appropriate technology tools in developing technology-based teacher support materials. Through the role and functions of computers especially the Internet, the student researchers and the advisers are able to design and develop various types of alternative delivery systems. These kinds of activities offer a remarkable learning experience for the education students as future mentors especially in the preparation of instructional materials. contents back next

The output of the group’s effort may serve as an educational research of the institution in providing effective and quality education. The lessons and evaluations presented in this module may also function as a supplementary reference for secondary teachers and students. GENALYN R. OBIAS Module Developer Module Developer MARIAN ANGELI A. PALMA contents back next

FOREWORD This Teacher’s Module entitled “ Methods of Teaching Mathematics ” is part of the requirements in Educational Technology 2 under the revised Education curriculum based on CHED Memorandum Order (CMO)-30, Series of 2004. Educational Technology 2 is a three (3)-unit course designed to introduce both traditional and innovative technologies to facilitate and foster meaningful and effective learning where students are expected to demonstrate a sound understanding of the nature, application and production of the various types of educational technologies. back next contents

The students are provided with guidance and assistance of selected faculty members of the College through the selection, production and utilization of appropriate technology tools in developing technology-based teacher support materials. Through the role and functions of computers especially the Internet, the student researchers and the advisers are able to design and develop various types of alternative delivery systems. These kinds of activities offer a remarkable learning experience for the education students as future mentors especially in the preparation of instructional materials. back next contents

The output of the group’s effort may serve as an educational research of the institution in providing effective and quality education. The lessons and evaluations presented in this module may also function as a supplementary reference for secondary teachers and students. FOR-IAN V. SANDOVAL Computer Instructor / Adviser Educational Technology 2 DELIA F. MERCADO Module Consultant LYDIA R. CHAVEZ Dean College of Education back next contents

ACKNOWLEDGEMENT The authors, who was given an incredible privilege to make this module, have a little congratulatory to themselves for such great performance that they gave. they would like to thank many dearest people who gave them the thrill of life and make this module possible. Dr. Corazon N. San Agustin, trained them and gives knowledge during Education 4A in making a module; For Ian V. Sandoval, approved the title of their module and give us the information that they needed in making our module; Prof. Lydia R. Chavez , Dean of education, approved and guide them in making this module; back next contents

Mr. Ricky R. Obias, for drawing the cover; Their parents , for understanding and motivation to not give up which adds to their determination in studying education; The people that is important to them, they want to say THANK YOU SO MUCH for being with. Most especially, they want to thank GOD who give His support and love to prove to ourselves that we can always do things possible. THE AUTHORS contents back next

INTRODUCTION Mathematics is a creative endeavor. It is a human activity which arises from experiences and becomes an integral part of culture and society, of everyday life and work. Teaching Mathematics effectively is quit hard to attain, sometimes the students found Mathematics as a boring subject they had, but as a student teacher, you must prepare yourself in these problems, you have to be flexible enough and creative enough to achieve your objectives and goals. Method is a procedure that one fallows in order to attain objectives, it stands for a specified course which serves as a guide in order “not to get lost on the way.” As a teacher someday, you must have a lots of methods know in teaching your subject in order to be more effective and more creative in the process of learning. contents back next

In this module entitled “METHODS OF TEACHING MATHEMATICS ”, will promote different methods in teaching mathematics, it will give ideas and information in different methods that can be used in teaching math. We are hoping that this module will help you to develop teaching mathematics in more easily and encouraging way. contents back next

GENERAL OBJECTIVES This module aims to present different methods in teaching Mathematics. Here are some expected outcomes to ensure the development of the student teachers. We intend to help you: learn more about some methods in order to undertake lessons attainable in Mathematics; 2. identify the different methods and its origin; 3. determine the ability to adopt in the method we present; 4. know the effectiveness of using methods in teaching; back contents next

5. introduce the quality and capability of different methods for the student teacher to balance the advantages and disadvantages of using these methods; 6. appreciate teaching Mathematics using different methods; 7. organize a creative and effective way of teaching; 8. relate to the present the capacity to teach math; 9. apply the different methods in teaching math; 10. get better acquainted with the methods; 11. construct simple mathematical activities that can apply these methods; 12. use the method on actual teaching; and 13. make lessons more motivated with the use of the methods. back contents next

TABLE OF CONTENTS Foreword VMGO Acknowledgement Introduction General Objectives Table of Contents Lesson 2 Teaching Basic Math Using Model method Lesson 1 Model Method Chapter I Teach Kids Math with Model Method Lesson 4 Socratic Method Chapter II Teaching Mathematics Using Socratic Method Lesson 3 Advantages and disadvantages of using Model Method Lesson 5 Application of Socratic Method in Teaching Mathematics back contents next

Lesson 6 Advantages and Disadvantages of Using Socratic Method Chapter III Teaching Mathematics Using Kumon Method Lesson 7 Kumon method Lesson 8 Strength of Kumon Method Lesson 9 Kumon Math Chapter IV Teaching Method that can be Use in Teaching Mathematics Lesson 11 Deductive Method Lesson 10 Lecture Method Lesson 12 Inductive Method Lesson 13 Project Method References back contents next

CHAPTER I TEACHING KIDS MATH WITH MODEL METHOD Objectives: At the end of the chapter, the students are expected to: 1. identify the Singapore model method and its origin; . give some examples that will practice specific math problems using model method; and . to improve teaching mathematics with the use of model method. contents back next

LESSON 1 THE MODEL METHOD Objectives: At the end of the lesson, the students are expected to: 1. determine where the model method originated; . know the way of learning mathematics in Singapore; and . measure the capacity to adopt model method. Origin of model method Singapore model math is a pedagogical strategy that was developed by a team of curriculum specialist in the Singapore ministry of education in the early 1980’s to address the issue of students having difficulty with word problems in early years of school. It has since become a distinguishing feature of the Singapore primary mathematics curriculum. contents back next

Using this method, students represent the information in the problem pictorially using bars to represent numbers. The model show explicitly the problem structure, the known and unknown quantities, and provides a visual tool that enable students to determine what operations to use to solve the word problems. Singapore Math is the generic name of the math curriculum or syllabus that is created by the creation ministry in Singapore for use in Singapore schools. Singapore’s math curriculum gained worldwide recognition when Singapore was ranked first in mathematics in the Third International Mathematics and Science study (TIMSS) in 1995 and Third International Mathematics and Science Study (TIMSS) in 1999. As more and more people around the world become fascinated with the outstanding success of Singapore students in mathematics, they began referring to Singapore’s math curriculum as simply Singapore math. Singaporean Mathematics contents back next

Singapore math is not confined to the primary school levels. The Singapore math curriculum extends beyond the primary school levels (first grade through sixth grade) to the secondary levels (seventh grade through tenth grade) and the junior college levels (eleventh grade through twelfth grade). The Singapore math curriculum for the primary school levels is also known as the Singapore Primary Math curriculum. The Singapore math curriculum for the secondary school levels is also known as the Singapore Secondary Math curriculum. Students in Singapore (and probably everywhere else) typically find word problems difficult due to various reasons: they are weak in the Mathematical language; they have limited understanding of the arithmetic operations; they are unable to relate the known to the unknowns when the problem structure is difficult to understand; and they are unable to analyze problem situations. contents back next

The model approach requires kids to draw rectangular boxes to represent part-whole relationships and math values (both known and unknown values) in the math problems. The word problems are typically designed to depict real-life situations such as grocery shopping and division of money. What is Model method? By drawing such boxes/blocks, they can visualize the math problems more clearly and are able to make tacit knowledge explicit. Word problem solving is a major part of the curriculum in Primary Mathematics in Singapore. This technique of model building is a visual way of picturing a situation. Instead of forming simultaneous equations and solving for the variables, model building involves using blocks or boxes to solve the problem. The power of using models can be best illustrated by problems, often involving fractions, ratios or percentages, which appear difficult but if models can be drawn to show the situation, the solution becomes clearer, sometimes even obvious. contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; __________________ Rating: __________________________ WORKSHEET NO.1 Direction: Read the questions properly and write the answer in the given line. Is the model method of the Singapore also applicable in teaching mathematics here in Philippines? Why? ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 2. As a student teacher, do you agree that solving mathematical problems are difficult to teach in the primary school? Why? _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ contents back next

Lesson 2 SOLVING MATHEMATICAL PROBLEM USING MODEL METHOD Objectives: At the end of the lesson, the students are expected to: . identify the ability of model method in teaching; . learn how to draw a diagram; and . apply model method in solving problems. The Part-Whole Concept The Part-Whole Concept is one of the three main concepts in the Model Method which all math models are derived from, the other two being the Comparison Concept and the Change Concept . The Singapore Math Primary Curriculum adopts a concrete-pictorial-abstract progressive approach to help pupils tackle seemingly difficult and challenging word problems. Mathematics Teachers in Singapore usually make use of concrete objects to allow students to make sense of the part-whole relationship between two or more parts. contents back next

Once the pupils can visualize the part-whole relationships, they will then move on to put these relationships in rectangular bars as pictorial representations of the math models concerned. To illustrate the part-whole concept, take a look at the following problem: Ann has 3 balls. Bob has 2 balls. How many balls do they have altogether? We can first give the child concrete objects, like 3 balls and another 2 balls, and let the child put the two groups of objects together to find the total. When they are comfortable with adding concrete objects, we can then proceed to teach them to draw pictures of the concrete objects within boxes to illustrate the equation 3 + 2 = 5 contents back next

After that, we can teach the kids to go on to draw the boxes without the objects. Eventually, the equation can be visualized as a whole made up of 2 parts and the pupils can easily see that to find the whole, they just need to add up the 2 parts. So, 3 + 2 = 5 contents back next

Therefore, they have 5 balls altogether. Hence, we can see that the relationship among the 3 quantities (the whole and 2 parts) can be summarized as follows: To find the whole given two parts, just add the two parts together: Part + Part = Whole To find one part when we are given the whole and the other part, just subtract the given part (or known part) from the whole. contents back next

Whole - Part = Part the Comparison Concept The Comparison Concept is one of the three main concepts in the Model Method which all math models are derived from, the other two being the Part-Whole Concept and the Change Concept . The Singapore Math Primary Curriculum adopts a concrete-pictorial-abstract progressive approach to help pupils tackle seemingly difficult and challenging word problems. Mathematics Teachers in Singapore usually make use of concrete objects to allow students to make sense of the comparison concept by comparing two or more quantities. Once the pupils can visualize how much one quantity is greater than or smaller than another quantity, they will then move on to put these relationships in rectangular bars as pictorial representations of the math models concerned. contents back next

To illustrate the comparison concept, take a look at the following problem: Peter has 5 pencils and 3 erasers. How many more pencils than erasers does he have? We can first give the child concrete objects, like 5 pencils and 3 erasers, and let the child put the two groups of objects side-by-side to match the 2 types of items, i.e., 1 pencils match with 1 eraser. Then, he will be able to see that there are 2 more pencils which cannot be matched with any erasers because he has run out of erasers to do that. When they are comfortable with comparing concrete objects, we can then proceed to teach them to draw pictures of the concrete objects within boxes to illustrate the equation 5 - 3 = 2 contents back next

Solving mathematical problems using Model method After that, we can teach the kids to go on to draw the boxes without the objects. contents back next

Eventually, the equation can be visualized as a comparison between the 2 quantities given in the question and the pupils can easily see that to find the difference; they just need to subtract the smaller quantity from the larger quantity. So, 5 - 3 = 2 Therefore, Peter has 2 more pencils than erasers. contents back next

Hence, we can see that the relationship among the larger quantity, the smaller quantity and the difference can be summarized as follows: To find the difference given two unequal quantities, just subtract the smaller quantity from the larger quantity: Larger Quantity - Smaller Quantity = Difference To find the larger quantity given the difference and the smaller quantity; just add the smaller quantity to the difference: Smaller Quantity + Difference = Larger Quantity contents back next

To find the smaller quantity given the difference and the larger quantity, just subtract the Difference from the larger quantity: Larger Quantity - Difference = Smaller Quantity The Change Concept The Change Concept is one of the three main concepts in the Model Method which all math models are derived from, the other two being the Part-Whole Concept and the Comparison Concept . You will, however, notice that the Change Concept is seemingly familiar when compared with the Part-Whole Concept and the Comparison Concept. The Singapore Math Primary Curriculum adopts a concrete-pictorial-abstract progressive approach to help pupils tackle seemingly difficult and challenging word problems. Mathematics Teachers in Singapore usually make use of concrete objects to allow students to make sense of the relationship between the new value of a quantity and its original value after an increase or decrease takes place. Once the pupils can visualize the change caused by the increase or decrease in quantity, they will then move on to put these relationships in rectangular bars as pictorial representations of the math models concerned contents back next

To illustrate the change concept involving an increase, take a look at the following problem: Peter has 3 marbles. Caleb gives Peter 1 more marble. How many marbles does Peter have now? We can first give the child concrete objects, like 3 marbles and another 1 marble, and let the child display the 3 marbles first and increase the total of the group by putting in 1 extra marble into the group to find the new value after the increase of 1 marble. When they are comfortable with increasing the original value to the new value with the concrete objects, we can then proceed to teach them to draw pictures of the concrete objects within boxes to illustrate the equation 3 + 1 = 4 contents back next

After that, we can teach the kids to go on to draw the boxes without the objects. Eventually, the equation can be visualized as an original value being increase by a certain value to obtain the new value. To find the new value after the increase for the above problem, they just need to add the "increase" to the original value. contents back next

So, 3 + 1 = 4 Therefore, Peter has 4 marbles now. Hence, we can see that the relationship among the original value, the increase and the new value can be summarized as follows: contents back next

To find the new value given the original value and the increase, just add the increase to the original value: Original Value + Increase = New Value to find the original value given the new value and the increase, just subtract the increase from the new value: New Value - Increase = Original Value to find the increase given the new value and the original value, just subtract the original value from the new value: New Value - Original Value = Increase contents back next

The Equal Concept The Equal Concept is derived from the Comparison Concept . It compares two or more fractions, decimals or percentages, etc that represent equal quantities. In this concept, we first draw a model to represent the first variable given and mark out the part of it that will be equal in quantity to a given part in the second variable represented by a second model. To illustrate this concept, consider the following question, 1/4 of A is equal to 1/3 of B. A is greater than B by 40. What is the value of A and B? Step 1: Draw a long bar to represent the whole of A. Divide the bar into 4 equal boxes and label 1 box as the equal part. contents back next

Step 2: Next, draw a box below the model of A to represent the part of B that is equal to 1/4 of A, i.e., 1/3 of B. Step 3: Since the first box of B drawn represents 1/3 of B, we will need to draw another 2 boxes to its right to represent the remaining 2/3 of B. contents back next

Step 4: Since A has 4 units and B has 3 units, the extra 1 unit of A must be equal to 40(given in question). 1 unit ----------> 40 3 units ----------> 3 X 40 = 120 4 units ----------> 4 X 40 = 160 Therefore, A is 120 and B is 160. 1 unit ----------> 40 3 units ----------> 3 X 40 = 120 4 units ----------> 4 X 40 = 160 Therefore, A is 120 and B is 160. contents back next

The Remainder Concept The Remainder Concept is derived from the Part-Whole Concept . Very often, questions which require the use of this Concept have the word "remainder" embedded in them. In this concept, we first draw a model to represent the whole and mark out the parts that were used or taken away from the whole. Then the "remainder part" of the model is subdivided according to the requirements given in the question. Eventually, all the known parts should be properly labeled with values and all the unknown parts should be divided equally so that we can work out their values. To illustrate this concept, consider the following question, Brandon gave 1/5 of his monthly salary to his mother. He gave ¾ of the remainder to his wife and saved the rest each month. He manage to save $400 every month. How much did he earn a month? contents back next

Step 1: Draw a long bar to represent the total of his salary Step 2: Divide the model into 5 equal parts and label 1 part as given to his mother. Step 3: Notice that there are 4 units left after giving 1/5 of his salary to his mother. Since he gave 3/4 of his salary to his wife, we label 3 of the remaining 4 units as given to his wife(3/4 is 3 out of 4 equal units). contents back next

Step 4: Since he had $400 left after that and there is only 1 unit of the model left, the last unit must be equal to $400. Hence,1 unit ----------> $400 5 units ----------> 5 X $400 = $2000 Therefore, Brandon earns $2000 a month. contents back next

The Constant Difference Concept The Constant Difference Concept is derived from the Comparison Concept . This concept is applicable when the problems deal with an equal quantity being transferred in or transferred out of the two variables concerned. This leaves the two variables with an equal increase or decrease in value. The unique feature in this concept lies in the fact that after the transfer in or transfer out of quantities, the difference between the two variables remains unchanged, hence the name "Constant Difference" Concept. To illustrate this concept, take a look at the following problems. (A) Equal Amount Transferred into 2 Variables Question: Ken had 14 pens and Ben had 2 pens. When they received an equal number of pens from their teacher, the ratio of Ken's pens to Ben's pens became 3:1. How many pens did each of them receive from their teacher? contents back next

Answer: For this question, we will work backwards. It is always easier to start drawing models where a multiples-relationship exist, i.e., a variable is a multiple of another variable. In this example, Ken is 3 times of Ben after the transfer in. Step 1: Since the ratio of Ken's pens to Ben's pens after the transfer in is 3:1, we draw 3 boxes to represent the number of units that Ken had and 1 box to represent the number of units that Ben had. Step 2: Since an equal amount was transferred in, we mark out an equal amount from both Ken's and Ben's model bars to show this amount. contents back next

Step 3: Next, we label the models with the number of pens they each have at first. Step 4: After all information have been put into the model, we can then mark out all the known parts and try to make all the unknown parts equal. contents back next

Step 5: From Ken's bar, we can see that, 2 units + 2 pens + 2 pens + 2 pens ----------> 14 pens 2 units + 6 pens ----------> 14 pens 2 units ----------> 14 pens - 6 pens = 8 pens 1 unit ----------> 8 pens / 2 units = 4 pen contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.2 Direction: Read the questions properly and write the answer in the given line. Can you adopt the model method in your teaching career someday? How? ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.3 Direction: Solve the mathematical problem using model method and indicate the steps of solving. Janet brought a box of cookies for her friends. If she gives each of them 4 cookies, she will have 4 cookies left. If she gives each of them 5 cookies she will need 5 more cookies. How many cookies are there in the box? contents back next

Lesson 3 ADVANTAGES and DISADVANTAGES OF USING BAR DIAGRAM Objectives: At the end of the lesson, the students are expected to: 1. show the importance of Model method; . share the idea of using Model method; and 3. prepare for the disadvantages of using bar diagram. Advantages of using bar diagrams: pictorial representation of abstract relationships fewer (and less abstract) rules to learn intuitive, easy to understand by students who have played with blocks flexible, applying in many situations diagrams eventually serve as a lead-in to algebra, giving students a way to visualize equations contents back next

Disadvantages of using bar diagrams: pictorial approach may not be intuitive to some students (or teachers) may not work in every situation takes plenty of practice to learn to use well In bar diagrams, quantities (both known and unknown) are represented by block-like rectangles. The student imagines moving these blocks around or cutting them into smaller pieces in order to find a useful relationship between the known and unknown quantities. In this way, the abstract puzzle of the word problem becomes a 2-D visualization puzzle: How can we fit these blocks together? For many students, this pictorial approach makes the problem much easier to understand. In the above problem, we would draw a rectangular bar to represent all the children. Then we would divide it into two parts, representing the boys and the girls: contents back next

As problems get more complex, the bar may be split into more than two parts. Also, the parts may be related to each other in ways that require a more involved diagram. However complicated the story, though, you usually begin by drawing a long bar to represent one whole thing and then dividing it into parts. Again, the student must learn some basic but important rules: The Whole Is the Sum of Its Parts All bar diagrams descend from one very basic diagram showing the inverse relationship between addition and subtraction: The whole is the sum of its parts . If you know the value of both parts, you can add them up to get the whole. If you know the whole total and one of the parts, you subtract the part you know in order to find the other part. In a picture contents back next

Simplify to a Single Unknown Part (called a “unit”) You cannot solve for two unknown numbers at once, so you must use the facts given in your problem and manipulate the blocks in your drawing until you can connect one unknown unit (or a group of same-size units) to a number. Once you find that single unknown unit, all the other quantities in your problem will fall into P contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.4 Direction: Read the questions properly and write the answer in the given line. 1. Now that you know the disadvantages of using mol method, do you still want to use it as a method of teaching mathematics? Why? ___________________________________________________________________________________________________________________________________________________________________________________________________ ___ 2. When you become a teacher someday, can you balance the advantages and disadvantages of teaching math? _________________________________________________________________________________________________________________________________________________________________________________ contents back next

CHAPTER II SOCRATIC METHOD: Teaching by asking instead by telling Objectives: At the end of the chapter, the students are expected to: . describe Socrates as an educator; . familiarize the student-teacher about the Socratic method; and . apply Socratic method in actual teaching. contents back next

Lesson 4 THE SOCRATIC METHOD Objectives: At the end of the lesson, the students are expected to: . define Socratic Method as a method in teaching; 2. impart the knowledge about Socrates; and . appreciate teaching math using Socratic Method. Socrates as an Educator Socrates as a Greek thinker and teacher . He was born in Athens about 470 B.C.and was put to death therein 399. He was married to Xanthippe, who was such a scold that her name is now used to mean “shrew”. For some time Socrates worked as a stoneman and sculptor. He spent the rest of his life thinking about philosophy and discussing it with practically everyone he met. contents back next

Socrates did not teach in the regular way. He held no classes and gave no lectures and wrote no books. He simply asked questions. “What is courage?” “Why do people do wrong actions?” when got an answer he asked more questions, like a cross-examination, until very often the other man admitted he could not give any answer. Socrates asked his students in order to make people think about ideas they took for granted. Some man admired this very much. They become fast friends of Socrates and join his philosophical discussion for many years. Others thought he was simply trying to destroy good old ideas about religion and morality without putting anything in their place. Some of the young man whom he knew well become traitors to their country and led a rebellion that overthrew the democratic government. The Athenians rose against them and killed tem. After democracy had been restored. Socrates was brought to trial. He was accused of introducing new goods to Athens and of corrupting young men’s minds. contents back next

Socrates did not take this seriously and would not ask for mercy. So he was condemned to die by drinking a cup of hemlock. Many people, then and later, thought the sentence was unjust because it denied freedom of speech. Others believed that he deserved to die because his pupils nearly destroyed the Athenian state. His most famous pupil, Plato, become a great philosopher and made Socrates the chief character in most of his books. Socrates and the Socratic Method: Socrates is known for the Socratic method ( elenchus ), Socratic irony , and the pursuit of knowledge. Socrates is famous for saying that he knows nothing and that the unexamined life is not worth living. The Socratic method involves asking a series of questions until a contradiction emerges invalidating the initial assumption. Socratic irony is the position that the inquisitor takes that he knows nothing while leading the questioning contents back next

Socratic Method Definition: Method of teaching pioneered by Socrates, the great Greek philosopher. The Method was a series of questions, by which Socrates made the people who answered the questions understand not only the point he was trying to make but also that they didn't know as much as they thought they did. An example of the Socratic Method is below: A single, consistent definition of the Socratic Method is not possible due to the diversity with which 'the method' has been used in history. There are many styles of question oriented dialogue that claim the name Socratic Method. However, just asking a lot of questions does not automatically constitute a use of the Socratic Method. Even in the dialogues of Plato, which are the most significant and detailed historical references to Socrates, there is not just one Socratic Method. The exact style and methodology of the Platonic Socrates changes significantly throughout the dialogues. If there is a 'classic' Socratic Method, this designation must refer to the style of the contents back next

Socratic Method found primarily in the early dialogues (also called the ‘Socratic Dialogues’) and some other dialogues of Plato. In these dialogues, Socrates claims to have no knowledge of even the most fundamental principles, such as justice, holiness, friendship or virtue. In the Socratic dialogues, Socrates only wants short answers that address very specific points and refuses to move on to more advanced or complicated topics until an adequate understanding of basic principles is achieved. This means that the conversation is often stuck in the attempt to answer what appears to be an unanswerable basic question. This image of Socrates' conversations, with their typical failure to find an answer, is the most widely recognized portrait of Socrates and his method. In the dialogues of Plato, the portrayal of Socrates and his method were diverse and ranged from the portrait of Socrates in the early dialogues to a richer diversity of conversational styles and ideas in latter dialogues. This diversity in the contents back next

dialogues was so great that Plato even decided to drop both Socrates and his method in some of his writing. In a later Platonic dialogue ‘The Laws’, there is still conversation but Socrates is replaced with ‘the stranger’ and his method is gone as well. Socrates and his method are most vividly seen in the early and middle dialogues. Two Styles of the Socratic Method In spite of their differences, both styles of the Socratic Method have some common aspects. Both can inspire people to increase their love of good questions. Both can draw people into a more thoughtful mode of thinking. The Modern Socratic Method can be used to good effect for leading a person to work out their own understanding of static knowledge such as mathematics . The Classic Socratic Method is a profoundly useful tool to facilitate improvements in critical thinking and to elevate the quality of human discourse regarding difficult and controversial issues. A contemporary example contents back next

of the Classic Socratic Method is the dialogue, The Moral Bankruptcy of Faith , where the Classic Socratic Method is used to demonstrate the necessity of caution when making overly broad statements about morality. The more difficult, ambiguous or controversial the issue, the more powerful the usefulness of the Classic Socratic Method will be in our conversations. This is because the need to think critically increases with the complexity and ambiguity of the issue or problem under discussion. Although some commonly shared level of problem solving and evaluative ability, which sometimes passes for critical thinking is used in our daily lives, the full and rich depth of the human capacity to think critically is much greater than ordinarily realized. Many people's ability to think with some measure of critical quality serves them fine in solving some practical problems. If, however, a problem has complex ethical dimensions or otherwise ambiguous qualities, the average ability to think critically is often not adequate. This inadequacy is especially evident when we are required to think critically about our own cherished beliefs and ideas. Although the contents back next

Classic Socratic Method is superior with regard to its impact on developing critical thinking, the Modern Socratic Method has a valuable influence on the development of critical thinking to the extent that it makes people comfortable questioning their own ideas. The good news about the Socratic Method is that some of its most powerful benefits are delivered to people in a way that does not require great philosophical prowess or teaching skill (Modern Socratic Method). A cup of open mindedness, a pinch of humble servility and a passion to explore makes up most of the recipe for putting the Modern Socratic Method to productive use. However, the most powerful aspect of the Socratic Method (the classic style) is very difficult to employ. Both styles of the Socratic Method are described below. contents back next

The Classic Socratic Method There are two phases in the Classic Socratic Method. I refer to the Classic Socratic Method as a Two-Phase Freestyle form of dialectic. The Modern Socratic Method is often constrained to a pre-designed set of questions that are known to generate a range of predictable answers and elicit knowable facts. The Classic Socratic Method is freestyle because, due to the nature of the questions, it cannot predict the responses to questions, anticipate the flow of the conversation or even know if a satisfactory answer is possible. The main portrait of how Socrates functioned in the classic style is in the early Dialogues of Plato (and some later dialogues). Plato wrote in the form of dialogues. In these dialogues Socrates would talk to people that had a reputation for having some knowledge of, or some interest in, the subject of the dialogue. In the classic style, Socrates would ask the primary question of the dialogue in the form of “What is X?”. (e.g. What is justice?) The respondents would answer. Socrates would then ask more questions and the contents back next

respondent’s answers would end up refuting the definition to the question "What is X?", which they had originally given. Once the respondent realized that the definition was not valid she would be asked again, “What is X?”. This process would often repeat until the end of the dialogue. With each new definition the respondent is subjected to more questions and continues to fail to define X. The conclusion of the dialogue would be an admission of failure to find a proper definition of X. Apparently this Socratic questioning had quite an effect on the respondents. In the Socratic dialogue called Meno , Socrates is asked by Meno if he believes that virtue can be taught. Meno was shocked and could scarcely believe it when Socrates tells him that he not only does not know if virtue can be taught, but does not understand the nature of virtue. Furthermore, Socrates tells Meno that he never knew anyone else who had an understanding of virtue. Meno’s contents back next

reluctance to believe Socrates never knew anyone who understood what virtue is was bason his belief that any grown and properly educated man would have some knowledge of virtue. Meno believed that he understood the nature of virtue. Meno is then exposed to Socratic questioning. Plato gives us a description of the effect this questioning had on Meno when Meno tells Socrates, “ O Socrates, I used to be told, before I knew you, that you were always doubting yourself and making others doubt; and now you are casting your spells over me, and I am simply getting bewitched and enchanted, and am at my wits' end. And if I may venture to make a jest upon you, you seem to me both in your appearance and in your power over others to be very like the flat torpedo fish, who torpifies (makes numb) those who come near him and touch him, as you have now torpified me, I think. For my soul and my tongue are really torpid, and I do not know how to answer you; and though I have contents back next

been delivered of an infinite variety of speeches about virtue before now, and to many persons-and very good ones they were, as I thought. At this moment I cannot even say what virtue is. And I think that you are very wise in not voyaging and going away from home, for if you did in other places as do in Athens, you would be cast into prison as a magician.” - from Meno Meno had been moved from a sense of security over his knowledge about virtue to the uncomfortable realization that he cannot even say what virtue is. With Meno’s words above we see the effect of the Classic Socratic Method. This effect has two main possibilities. Either a person will be inspired to better and more vigorous thinking about a question or they will get discouraged by having their perspective challenged. contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.5 Direction: Read the questions properly and write the answer in the given line. 1. Describe Socrates base on what you have been read. ______________________________________________________________________________________________________________________ 2. Why some people think that questions and answers seemed to bizarre or “tricky” instead of logical? ______________________________________________________________________________________________________________________ 3. Is the Socratic Method guide the students to clear thinking? How? ____________________________________________________________________________________________________________________ next contents back

Lesson 5 APPLICATION OF SOCRATIC METHOD Objectives: At the end of the lesson, the students are expected to: determine the use of Socratic Method in teaching Mathematics; 2. prove the Socratic Method as an effective Method; and 3. apply Socratic Method in teaching. Socratic Teaching The oldest, and still the most powerful, teaching tactic for fostering critical thinking is Socratic teaching. In Socratic teaching we focus on giving students questions, not answers. We model an inquiring, probing mind by continually probing into the subject with questions. Fortunately, the abilities we gain by focusing on the next contents back

elements of reasoning in a disciplined and self-assessing way, and the logical relationships that result from such disciplined thought, prepare us for Socratic questioning. Thankfully, there is a predictable set of relationships that hold for all subjects and disciplines. This is given in the general logic of reasoning, since every subject has been developed by those who had: * shared goals and objectives (which defined the subject focus) * shared questions and problems (whose solution they pursued) * shared information and data (which they used as an empirical basis) * shared modes of interpreting or judging that information * shared specialized concepts and ideas (which they used to help them organize their data) shared key assumptions (that gave them a basis from which to collectively begin) a shared point of view (which enabled them to pursue common goals from a common framework) next contents back

Each of the elements represents a dimension into which one can delve in questioning a person. We can question goals and purposes. We can probe into the nature of the question, problem, or issue that is on the floor. We caninquire into whether or not we have relevant data and information. We can consider alternative interpretations of the data and information. We can analyze key concepts and ideas. We can question assumptions being made. We can ask students to trace out the implications and consequences of what they are saying. We can consider alternative points of view. All of these, and more, are the proper focus of the Socratic questioner. As a tactic and approach, Socratic questioning is a highly disciplined process. The Socratic questioner acts as the logical equivalent of the inner critical voice which the mind develops when it develops critical thinking abilities. The contributions from the members of the class are like so manythoughts in the mind. All of the thoughts must be dealt with and they must be dealt with carefully and fairly. By following up all answers with further questions, and by selecting questions which advance the contents back next

discussion, the Socratic questioner forces the class to think in a disciplined,intellectually responsible manner, while yet continually aiding the students by posing facilitating questions. A Socratic questioner should: a) keep the discussion focused b) keep the discussion intellectually responsible c) stimulate the discussion with probing questions d) periodically summarize what has and what has not been dealt with and/or resolved e) draw as many students as possible into the discussion. Here is an example of using Socratic Method in actual teaching. contents back next

I am the subject of the experiment, not you. I want to see whether I can teach you a whole new kind of arithmetic only by asking you questions. I won't be allowed to tell you anything about it, just ask you things. When you think you know an answer, just call it out. You won't need to raise your hands and wait for me to call on you; that takes too long." [This took them a while to adapt to. They kept raising their hands; though after a while they simply called out the answers while raising their hands.] Here we go. 1) "How many is this?" [I held up ten fingers.] TEN 2) "Who can write that on the board?" [virtually all hands up; I toss the chalk to one kid and indicate for her to come up and do it]. She writes 10 contents back next

3) Who can write ten another way? [They hesitate than some hands go up. I toss the chalk to another kid.] 4) Another way? 5) Another way? 2 x 5 [inspired by the last idea] 6) That's very good, but there are lots of things that equal ten, right? [student nods agreement], so I'd rather not get into combinations that equal ten, but just things that represent or sort of mean ten. That will keep us from having a whole bunch of the same kind of thing. Anybody else? TEN contents back next

7) One more? X [Roman numeral] 8) [I point to the word "ten"]. What is this? THE WORD TEN 9) What are written words made up of? LETTERS 10) How many letters are there in the English alphabet? 26 11) How many words can you make out of them? ZILLIONS 12) [Pointing to the number "10"] What is this way of writing numbers made up of? NUMERALS contents back next

13) How many numerals are there? NINE / TEN 14) Which, nine or ten? TEN 15) Starting with zero, what are they? [They call out, I write them in the following way.] 0 1 2 3 4 5 6 7 8 9 contents back next

16) How many numbers can you make out of these numerals? MEGA-ZILLIONS, INFINITE, LOTS 17) How come we have ten numerals? Could it be because we have 10 fingers? COULD BE 18) What if we were aliens with only two fingers? How many numerals might we have? 2 19) How many numbers could we write out of 2 numerals? NOT MANY / [one kid:] THERE WOULD BE A PROBLEM 20) What problem? THEY COULDN'T DO THIS [he holds up seven fingers] contents back next

21) [This strikes me as a very quick, intelligent insight I did not expect so suddenly.] But how can you do fifty five? [he flashes five fingers for an instant and then flashes them again] 22) How does someone know that is not ten? [I am not really happy with my question here but I don't want to get side-tracked by how to logically try to sign numbers without an established convention. I like that he sees the problem and has announced it, though he did it with fingers instead of words, which complicates the issue in a way. When he ponders my question for a second with a "hmmm", I think he sees the problem and I move on, saying...] 23) Well, let's see what they could do. Here's the numerals you wrote down [pointing to the column from 0 to 9] for our ten numerals. If we only have two numerals and do it like this, what numerals would we have. 0, 1 contents back next

24) Okay, what can we write as we count? [I write as they call out answers.] 0 ZERO 1 ONE [silence] 25) Is that it? What do we do on this planet when we run out of numerals at 9? WRITE DOWN "ONE, ZERO" 26) Why? [almost in unison] I DON'T KNOW; THAT'S JUST THE WAY YOU WRITE "TEN" 27) You have more than one numeral here and you have already used these numerals; how can you use them again? WE PUT THE 1 IN A DIFFERENT COLUMN 28) What do you call that column you put it in? TENS 29) Why do you call it that? DON'T KNOW contents back next

30) Well, what does this 1 and this 0 mean when written in these columns? 1 TEN AND NO ONES 31) But why is this a ten? Why is this [pointing] the ten's column? DON'T KNOW; IT JUST IS! 32) I'll bet there's a reason. What was the first number that needed a new column for you to be able to write it? TEN 33) Could that be why it is called the ten's column?! What is the first number that needs the next column? 100 34) And what column is that? HUNDREDS 35) After you write 19, what do you have to change to write down 20? 9 to a 0 and 1 to a 2 contents back next

36) Meaning then 2 tens and no ones, right, because 2 tens are ___? TWENTY 37) First number that needs a fourth column? ONE THOUSAND 38) What column is that? THOUSANDS 39) Okay, let's go back to our two-fingered aliens arithmetic. We have 0 zero 1 one. What would we do to write "two" if we did the same thing we do over here [tens] to write the next number after you run out of numerals? START ANOTHER COLUMN 40) What should we call it? TWO'S COLUMN? contents back next

41) Right! Because the first number we need it for is ___? TWO 42) So what do we put in the two's column? How many two's are there in two? 1 43) And how many one's extra? ZERO 44) So then two looks like this: [pointing to "10"], right? RIGHT, BUT THAT SURE LOOKS LIKE TEN. 45) No, only to you guys, because you were taught it wrong [grin] -- to the aliens it is two. They learn it that way in pre-school just as you learn to call one, zero [pointing to "10"] "ten". But it's not really ten, right? It's two -- if you only had two fingers. How long does it take a little kid in pre-school to learn to read numbers, especially numbers with more than one numeral or column? TAKES A WHILE contents back next

46) Is there anything obvious about calling "one, zero" "ten" or do you have to be taught to call it "ten" instead of "one, zero"? HAVE TO BE TAUGHT IT 47) Ok, I'm teaching you different. What is "1, 0" here? TWO 48) Hard to see it that way, though, right? RIGHT 49) Try to get used to it; the alien children do. What number comes next? THREE 50) How do we write it with our numerals? We need one "TWO" and a "ONE" [I write down 11 for them] So we have 0 zero 1 one 10 two 11 three contents back next

51) Uh oh, now we're out of numerals again. How do we get to four? START A NEW COLUMN! 52) Call it what? THE FOUR'S COLUMN 53) Call it out to me; what do I write? ONE, ZERO, ZERO [I write "100 four" under the other numbers] 54) Next? ONE, ZERO, ONE I write "101 five" 55) Now let's add one more to it to get six. But be careful. [I point to the 1 in the one's column and ask] If we add 1 to 1, we can't write "2", we can only write zero in this column, so we need to carry ____? ONE 56) And we get? ONE, ONE, ZERO contents back next

57) Why is this six? What is it made of? [I point to columns, which I had been labeling at the top with the word "one", "two", and "four" as they had called out the names of them.] a "FOUR" and a "TWO" 58) Which is ____? SIX 59) Next? Seven? ONE, ONE, ONE I write "111 seven" 60) Out of numerals again. Eight? NEW COLUMN; ONE, ZERO, ZERO, ZERO I write "1000 eight" [We do a couple more and I continue to write them one under the other with the word next to each number, so we have:] contents back next

0 zero 1 one 10 two 11 three 100 four 101 five 110 six 111 seven 1000 eight 1001 nine 1010 ten 61) So now, how many numbers do you think you can write with a one and a zero? MEGA-ZILLIONS ALSO/ ALL OF THEM 62) Now, let's look at something. [Point to Roman numeral X that one kid had written on the board.] Could you easily multiply Roman numerals? Like MCXVII times LXXV? NO contents back next

63) Let's see what happens if we try to multiply in alien here. Let's try two times three and you multiply just like you do in tens [in the "traditional" American style of writing out multiplication]. 10 two x 11 times three They call out the "one, zero" for just below the line, and "one, zero, zero" for just below that and so I write: 10 two x 11 times three 10 100 110 64) Ok, look on the list of numbers, up here [pointing to the "chart" where I have written down the numbers in numeral and word form] what is 110? SIX contents back next

65) And how much is two times three in real life? SIX 66) So alien arithmetic works just as well as your arithmetic, huh? LOOKS LIKE IT 67) Even easier, right, because you just have to multiply or add zeroes and ones, which is easy, right? YES! 68) There, now you know how to do it. Of course, until you get used to reading numbers this way, you need your chart, because it is hard to read something like "10011001011" in alien, right? RIGHT 69) So who uses this stuff? NOBODY/ ALIENS 70) No, I think you guys use this stuff every day. When do you use it? NO WE DON'T contents back next

71) Yes you do. Any ideas where? NO 72) [I walk over to the light switch and, pointing to it, ask:] What is this? A SWITCH 73) [I flip it off and on a few times.] How many positions does it have? TWO 74) What could you call these positions? ON AND OFF/ UP AND DOWN 75) If you were going to give them numbers what would you call them? ONE AND TWO/ [one student] OH!! ZERO AND ONE! [other kids then:] OH, YEAH! contents back next

76) You got that right. I am going to end my experiment part here and just tell you this last part. Computers and calculators have lots of circuits through essentially on/off switches, where one way represents 0 and the other way, 1. Electricity can go through these switches really fast and flip them on or off, depending on the calculation you are doing. Then, at the end, it translates the strings of zeroes and ones back into numbers or letters, so we humans, who can't read long strings of zeroes and ones very well can know what the answers are. [at this point one of the kid's in the back yelled out, OH! NEEEAT!!] I don't know exactly how these circuits work; so if your teacher ever gets some electronics engineer to come into talk to you, I want you to ask him what kind of circuit makes multiplication or alphabetical order, and so on. And I want you to invite me to sit in on the class with you. contents back next

Now, I have to tell you guys, I think you were leading me on about not knowing any of this stuff. You knew it all before we started, because I didn't tell you anything about this -- which by the way is called "binary arithmetic", "bi" meaning two like in "bicycle". I just asked you questions and you knew all the answers. You've studied this before, haven't you? NO, WE HAVEN'T. REALLY. Then how did you do this? You must be amazing. By the way, some of you may want to try it with other sets of numerals. You might try three numerals 0, 1, and 2. Or five numerals. Or you might even try twelve 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ~, and ^ -- see, you have to make up two new numerals to do twelve, because we are used to only ten. Then you can check your system by doing multiplication or addition, etc. Good luck. After the part about John Glenn, the whole class took only 25 minutes. contents back next

Their teacher told me later that after I left the children talked about it until it was time to go home. contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.6 Direction: Read the questions properly and write the answer in the given line. 1. Do you think that the study using Socratic Method is a student-centered? Why? ____________________________________________________________________________________________________________________ 2. How important logical thinking is, when it comes to Mathematics? __________________________________________________________________________________________________________________ 3. Do you agree that logical questions help the students to developed thinking skill? How? __________________________________________________________________________________________________________________ contents back next

Lesson 6 ADVANTAGES AND DISADVANTAGES OF USING SOCRATIC METHOD Objectives: At the end of the lesson, the students are expected to: . know the capacity and limitations of Socratic Method; . show the good benefits of using Socratic Method; and . compare the Method in other Method. Advantages of using SM The Socratic Method of teaching has been used by schools and teachers for many generations. However, it is often more used in higher education, which comes as a disadvantage to young students. Young students should be taught by the Socratic Method because it encourages critical thinking , in-class participation, and other social skills that students will need to succeed at any level of education contents back next

The Socratic Method, first developed by Socrates, is an easy-to-understand principle. It involves discussion; by discussing a certain topic or book or thought, students can actively engage with their knowledge instead of simply memorizing or retaining it. Students can also exchange opinions and ideas, and develop excellent speaking and communication skills. The Socratic Method can be used in virtually every subject, although in slightly different ways. For humanities teachers, the Socratic Method is easy to employ. English, social science, and philosophy teachers will be able to use the Socratic Method without even changing their curriculum. For example, once your English class has finished reading a book, place the students' desks in a circular shape and begin discussing the theme, plot, or overall influences of that book. All of the ideas which are discussed and evaluated can be used later on in presentations, essays, summaries, etc. contents back next

Science and mathematics-based teachers can also use the Socratic Method on alternating days with their regular lesson plans. For example, a calculus teacher may be able to lecture on Mondays, Wednesdays, and Fridays, and hold Socratic Method seminars on Tuesdays and Thursdays, in which the students may discuss theorems and principles of calculus. By presenting a subject like calculus in two different ways, calculus teachers will be able to really instill the value of the subject in their students. Socratic seminars, however, must be "refereed" constantly. If you do not monitor your students' discussion, they might go off-topic and begin debating about personal values or opinions that may be influenced by race or political or religious backgrounds. These conversations should be avoided or, at the very least, limited. These discussions usually become very personal, and the discussion can only go in circles. If you find that your students are in this situation, change the topic immediately. contents back

The Socratic Method is a fun yet educational way to teach your students how to make use of their knowledge. The Socratic Method also teaches students how to think critically, accept others' opinions or viewpoints, and apply their knowledge to the real world and to other forms of knowledge. Disadvantages of using SM Assume Everyone Knows Nothing. Whether you are a Science teacher, English, or even History teacher, you want to be well researched and versed in your own studies as an educator. This way you have some knowledge in which to pose questions to your pupils. If you do not know your own material, then you will not have the resources to formulate a question. Yet while in class, you should ask a question without assuming that you know the answer. For example, Wikipedia uses the phrase- "Courage is endurance of the soul." At this point there should be opposing viewpoints to "stimulate thinking and to illuminate ideas" contents back next

Don't Bite the Agreement So Early. Even if a student readily agrees with the statement you proposed, such as "Courage is endurance of the soul" should you stop the discussion? No. This method is used plenty in law schools of America. Students are asked to review cases and form their own statements about it in their own words. The same principle should apply in high school and beyond. At this point you would probably call on someone else and both of you would begin to "argue". At this juncture, you would probably answer a question with a question. This is the true form of the Socratic Method of teaching. It is Imperative to Know You Can Be Wrong. Using the Socratic way of conducting class, means that you as the teacher can be proven wrong by a student, or have some fallacies in your responses. The rule of the game is to know that no one is better than anyone. Socrates was called wise by the Oracle and others because he was wise enough to understand that even he knew nothing contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.7 Direction: Read the questions properly and write the answer in the given line. 1. How can you manage a lesson if your students are not interested with the topic? ____________________________________________________________________________________________________________________ __________________________________________________________ 2. What are the skills that are developed when teaching mathematics using Socratic Method? ______________________________________________________________________________________________________________________________________________________________________________ contents back next

Chapter III KUMON METHOD Objectives: At the end of the chapter, the students are expected to: determine what kumon method is; distinguish why kumon method exist; and appreciate the effectiveness of this method. contents back next

Lesson 7 KUMON METHOD Objectives: At the end of the lesson, the students are expected to: understand and explain the kumon method; and explain the vision and mission of kumon method. Kumon method was founded in 1958 by Mr. Toru Kumon , who was a senior high school math teacher. When he taught his son math, he found the traditional text book so boring that it hardly arouses any interest in children. Mr. Toru believes that children’s potential in math could be totally developed a self-study and independent learning method supported by a gradual progress which is designed according to their own abilities. contents back next

Kumon is a unique method of study which aims to give each and every child skills that lasts a lifetime. Focusing on the vital subjects of math and English, you can help the children to develop their learning skills and exceed expectations with the worksheets and study center-based programs. At Kumon, you are passionate about the child’s development. Whatever their age and ability, the programs helps the child: Master math and English skills. Develop excellent concentration and study habits. Build lasting confidence and self belief. Improve their all-around academic achievements. contents back next

Kumon’s Mission By discovering the potential of each individual and developing his/her ability to the maximum, aim to foster sound, capable people and thus contribute to the global community. Kumon’s Vision “ World peace through education.” Kumon’s vision is really quite simple, and yet incredibly humanistic. The vision of world peace through education will be realized by the fostering of as many capable individuals as possible. Such individuals will progress the skills to overcome hardships and difficulties using their own innate abilities. By doing so, we believe that we can realize our vision, world peace, for a well-educated community made up of capable individuals will be a peaceful community. contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.8 Direction: Read the questions properly and write the answer in the given line. 1. Are you agree that through the use of kumon method they can reach their mission and vision? Explain. __________________________________________________________________________________________________________________ 2. What can you say about kumon method? __________________________________________________________________________________________________________________ contents back next

Lesson 8 STRENGTHS OF KUMON METHOD Objectives: At the end of the lesson, the students are expected to: . identify and explain the strengths of kumon method; and 2. appreciate the importance of its strengths. 1. Individual Learning The path of study is tailored according to student’s abilities. The Starting Point, daily workload, projection, amount of repetition, rate of advancement, are all designed with each individual in mind. No students are alike. Those who exhibit mastery can advance forth; those who need reinforcement would be assigned with repeated practice and advance forth only when ready. contents back next

2. Easy Starting Point You may be surprised at the easy starting point set for students. The easy starting point is set so that students can begin their Kumon experience with success, attaining 100% with each set and staring to build confidence in mathematics. It is also important for other reasons. Whenever beginning any new ability such as joggling or swimming, it is always best to take it easy first to ease into the new routine. In Kumon, starting at an easy level facilitates the development of concentration skills. 3. Self Learning Kumon is designed in minute steps. This gives students the ability to learn by themselves. Each set of work accomplish is a step towards the next. The worksheets, used in the correct way, will give the students the ability to solve problems. If learning will bring about a greater understanding of the questions and mathematical process. Examples are given when students start something new. next contents back

4.Daily Study Kumon is a daily program. The students spend only 10-30 minutes a day to do the worksheets. This will instill in them good study habits and self discipline. Constancy and continuity are great assets for the future education as well as self development. To skip days or to allow homework to pile up is to invite problems or trouble. The students may forget what they have learnt and they would suffer or be discouraged by the pile of unfinished homework. 5. Repetition Repetitions are one of the reasons for Kumon’s success. It gives students adequate time to consolidate and master each area covered in the Kumon program. So it is quite normal and natural for students to be asked to repeat worksheets that have already been completed. Excellence can always be achieved with adequate consolidation. With repetition each level will be easier if the previous level is consolidated. next contents back

6. Standard Completion (STC) Each set (10 pages back and front) of worksheets has a specific standard time to be completed. This time has been thoroughly tested and is well within the competent student’s reach. It is not a race with the clock. If the STC is reach with the student working at a normal face and writing neatly, you will see that these worksheets have been mastered. If the student cannot complete the worksheets within SCT then it is better to revise this work. The student who doesn’t reviser will eventually progress at a MUCH LOWER rate. contents back next

Direction: Read the questions properly and write the answer in the given line. Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.9 1. Enumerate and discuss in your own words the strengths of kumon method; _________________________________________________________________________________________________________________________________________________________________________________ 2. Are the strengths that have mentioned effective? Why? _________________________________________________________________________________________________________________________________________________________________________________ contents back next

Lesson 9 KUMON MATH Objectives: At the end of the lesson, the students are expected to: 1. identify the difference between kumon math and tutoring; 2. know the disadvantages of kumon math and how it become disadvantages; and 3. describe the role of the worksheets in kumon. Kumon Math follows the Kumon Method, a series of graded math worksheets that students work through independently, to master the basic skills of mathematics. There are more than twenty graded levels within the Kumon Math’s program starting from toddler to college levels and students work through them systematically, repeatingly and repeatingly each contents next back

set as necessary, until they have mastered the skill being taught in that series of worksheets. Each student is tested at the start of their program, and begins at a level that is easy for them; what kumon calls a “comfortable starting point.” This helps to increase the students’ confidence and motivation whist reinforcing basic skills. One of the key features of the Kumon Method is that no calculators are used. As a result students become fast and accurate at basic arithmetic as more advanced mathematics. Difference between Kumon Math and Tutoring Kumon does not offer children individual tutoring. The majority of the math worksheets are done at home and marked by parents. Instead new concepts are introduced at the start of a new worksheet booklet, providing a simple example, and the student contents back next

then begins to attempt questions immediately. The work is graded and introduced in such small steps that there appears to be a natural progression. This avoids students becoming discouraged and losing confidence. Once or twice a week the students attends a kumon center where a supervisor checks the completed work and assigns new worksheets. They also administer any assessment tests that may be required. Students have to gain complete mastery of a subject before they are allowed to more on to the next level. This is judged not only by the accuracy of their work but by the time in which it is completed. One of the disadvantages of is that it allows student to progress at their own pace. Students are not required to study at their grade level, rather, they progress to the level at which they are comfortable. Often students may be studying 2- 3 years above their school grade level. contents back next

Many students of Kumon Math report improving their class positions as well as greatly increasing their confidence in mathematics. Disadvantages of Kumon Math Whilst there is no question that for many children kumon works at improving their confidence in their own abilities, there are also disadvantages of the system. One of these is that many students complain that the worksheets are boring. Often students are required to repeat worksheets as many as 6 or 8 times. For some students it is hard to maintain their motivation for the program. Another disadvantage is the time involved. Kumon study usually requires around 20 minutes of focused study each day. For some busty families this proves a large stumbling block to following the program. Whilst parents are not required to actively teach their children many centers ask that they mark and correct the workbooks and also that students times to complete worksheets are recorded. contents back next

Although Kumon covers key elements of mathematics at the primary school age it does not attempt to cover the whole curriculum until high school age. This means that for junior school pupils only 15-20% of the mathematics required is covered. Kumon focuses on the key skills of addition, subtraction, multiplication, and division in the early stages. Another disadvantage is cost. In some countries the program costs as much as S100 per subject a month. For many facilities this can be prohibitively expensive. Alternatives to the Kumon Program Although the Kumon Program is fairly unique there are several websites that offer online printable mathematics worksheets. Many of those websites offer them by school year and by concept to be learnt. These an provide an alternative for a dedicated parent. contents back next

Kumon themselves also published a huge range of kumon worksheets for children to complete at home without actually following the kumon program. Whilst these are not comprehensive as the kumon program itself they are substantially cheaper and are a good substitute foe someone looking for a most effective alternative. Worksheets “ Small steps lead to greater success.” Significant jumps from learning level to learning level are discourage. This will only cause difficulties for students and impair learning. The materials used by the kumon method have been constructed in a manner in which students’ progress in small steps through the learning process. At all times, each student is learning at a level that is just right for him o her. Worksheets are constructed in such a way that students never encounter problems with which they are unfamiliar. Sample problems, that include many examples, are always provided when moving into new areas of learning, assisting students in learning these materials on their own. As a result, students can affectively learn by themselves. contents back next

The mathematics program consists of 4,540 worksheets with 23 levels, from level 7A through Q, with these levels extending in difficulty directly proportionate to the alphabetical listings, with 7A being the easiest and Q the most difficult. The worksheets focus on the development of strong calculation skills and aim to assist students in advancing by them as directly as possible to high school mathematics by avoiding all irrelevant concepts. This is the prime feature of the kumon worksheets. Kumon consistently checks to assume that no problem areas exist within the materials that would cause hindrances to the students. All worksheets are reviewed routinely, and continuous revisions are made based upon feedback from students and instructors. In every way, this is a learning method that continually places the students at the core of the learning process. contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.10 Direction: Read the questions properly and write the answer in the given line. 1. As a future teacher, could you recommend this method to use in teaching mathematics? Why? _________________________________________________________________________________________________________________________________________________________________________________ 2. What can you say about the use of worksheets in this method? _________________________________________________________________________________________________________________________________________________________________________________ contents back next

Chapter IV TEACHING METHODS THAT CAN BE USED IN TEACHING MATHEMATICS Objectives: At the end of the chapter, the students are expected to: determine what teaching methods are can be used in teaching mathematics; and demonstrate and explain how to apply teaching methods presented. contents back next

Lesson 10 LECTURE METHOD Objectives: At the end of the lesson, the students are expected to: discuss how to prepare lecture for teaching; describe how to use lecture method; and understand when to use lecture method. Other Factors . Teacher personality weighs a great deal in the effectiveness of the lecture. A good voice, good pronunciation and enunciation, proper facial expressions, gestures and manners of lecturing should be taken into great consideration. The style of delivery and the manner of lecturing should be appropriate to the subject matter and comfortable for the teacher. The teacher should talk in a poised, animated, conversational style and smile in the right places. He should look at students and talk to them directly. contents back next

I the lecture, the teacher has a great responsibility to guide the thinking of the students and so he must make himself intelligible to them. Unlike other methods where motivations can come from subsequent activities, in the lecture, student interest depends largely on the teacher. Getting the attention is another factor the teacher must master. Various aids may be utilized to master this problem, they are: introducing visual aids, varying the pause and tempo of his presentation, changing his voice, by using novelty, surprise and illustrations. Getting and holding attention also depends upon elimination of distractions. Comprehension by the class is the measure of success of the lecture to insure comprehension, two approaches may be used. The first is to have repetition or approach from another angle of thought. The second is to remove the causes of difficulty by using verbal and concrete illustrations. contents back next

Advantages of the Lecture Method Gives the teacher the chance to expose students to all kinds of material. Allows the teacher to precisely determine the aims, content, organization, pace and direction of a presentation. Can be used to amuse interest in a subject. Can complete and clarify text material. Complements certain individual learning preferences. Facilitates large-class communication. Disadvantages of the Lecture Method Places student in a passive rather than an active role, which hinders learning. Encourages one-way communication; therefore, the learner must make a conscious problems and student understanding of content without verbal feedback. Requires a considerable amount of unguided student in time outside of the classroom to enable understanding and long term retention of content. contents back next

Preparing the Teaching Lecture 1. Planning: The following four steps are followed in the planning phase of preparation: Establishing the objective and desired outcomes; Researching the subject; Organizing the material; and Planning productive classroom activities. In all stages of preparing for the teaching lecture, the teacher should support any point to be covered with meaningful examples, comparisms, statistics or testimony. While developing the lesson, the teacher also should strongly consider the use of examples and personal experiences related to the subject of the lesson. 2. Rehearsing: After completing the preliminary planning and writing of the lesson plan, the teacher should rehearse the lecture to build self-confidence. It helps to smooth out the use notes, visual aids and other instructional devices. contents back next

Delivering a Lecture 1. Suitable Language: In the teaching lecture, simple rather than complex words should be used whenever possible. The teacher should not we substandard English. If the subject matter includes technical terms, the teacher should clearly define each one so that no student is in doubt its meaning. Whenever possible, the teacher should use specific rather than general words. 2. Tone and Pace: Another way the teacher can add life to the lecture is to vary his or her tone of voice and pace of speaking. In addition, using sentences of different length also helps. To ensure clarity and variety, the teacher should normally use sentences of short and medium length. 3. Use of Notes: For a teacher notes are a must because they help keep the lecture on track. The teacher should use them modestly and should make no effort to hide them from the students. Notes may be written legibly or typed and they should be placed where they can be consulted easily. contents back next

Strengths The lecture may serve as a very effective means of amusing appreciation. A work of art, a musical composition or a literary selection may be better appreciated if preceded by a lecture that explains its meaning and the circumstance of its creation. The lecture may also serve to motivate a study; for instance, the life of Edison: by telling the historical or biographical background, the teacher may put the class in the right emotional tone. New topics may be introduced by a lecture. The teacher usually gives a short lecture at the beginning of a unit, problem or a contract. The lecture trains students to listen, they listen to radio, they listen to the TV and at the movies. They have to listen to many situations in life. Life includes relating incidents, telling stories, explanations, etc. which are forms of the lecture. contents back next

Weaknesses The greatest objection is that it violates the principle of “learning by doing.” This is the reason it is not often used in the elementary and high school. Moreover, it fosters a passive attitude in the class. Where students are immature, a sustained lecture will be just a waste of time. It will be a waste of time if what is lectured can be founded in the text, or if the material is available to the student, or if the teacher lectures on what he assigned. The lecture may not hold the attention of the class for various reasons, such as: (1) the teacher may not know the techniques of lecturing; (2) the teacher may over-use the lecture; (3) the listeners may be too young. The lecture may also be ineffective as a method because (1) the students may not be able to distinguish the important from the not-so-important points; (3) students may not know how to analyze and summarize. contents back next

Other disadvantages that may be mentioned are: students lack the opportunity to study in advance. The learner becomes a mere recipient instead of a thinker. Merely telling facts does not guarantee that these will be thought about, learned and used. The material may not be remembered or applied. There is lack of opportunity of discussion and expression. During the lecture, there is no way of finding out whether the class is getting the right ideas or the wrong ones. contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.11 Direction: Read the questions properly and write the answer in the given line. 1. How important in this method the ability of a teacher to communicate effectively? ____________________________________________________________________________________________________________________ 2. Most of the teachers use this method in teaching, is it because of its effectiveness? ____________________________________________________________________________________________________________________ 3. Do you think this method is effective in teaching mathematics? Explain your answer. __________________________________________________________________________________________________________________ contents back next

Lesson 11 DEDUCTIVE METHOD Objectives: At the end of the lesson, the students are expected to: 1. discuss the strengths and weaknesses of deductive method;. 2. describe the deductive method; and 3. enumerate and explain the steps of deductive method. Deduction is the process of solving a problem by applying to the problem or difficulty a generalization already formed. It is the process of thought starting from general going to particular. The deductive procedure starts with a rule that is applied to specific cases for the purpose of testing its validity, illustrating or further developing it, or solving the problem to which it applies. The Deductive Methods is used for the following purposes: to teach students to delay judgment until truth is proven and not to judge even in the face of seeming certainly: to master difficulties by utilizing truth established by others and: to remedy or overcome the tendency to jump to conclusions at once. contents back next

Deduction may be anticipatory deduction which forecast details that will be found in a particular situation, or explanatory deduction which connects facts at hand with principles that interpret them. This type is often used in the classroom when the teacher asks for the principle that explains this or that phenomenon. Most textbooks teaching makes use of this method too. The principle or explain it. From experience, this method works well with comparatively slow moving groups of students. Steps of the Deductive Method Statement of the Problem. The problem should be motivating and should arouse a desire to solve it. As much as possible, it should be related to a life situation, should be real, vital and within the ability and maturity of the student. Generalization . Too or more generalizations may be recalled. One of these will be the solution to the problem. Inference. This is choosing from among the generalizations the one that will fit the problem. contents back next

Verification . This is trying out and securing the successful generalization. It is determining the validity of the inference by consulting accepted authorities such as the teacher, the textbook, the encyclopedia, or other resource persons. Merits of Deductive Methods Deduction Method is short time-saving. It takes little time to solve the problem by predetermined formulae. In the Deductive Method, the teacher’s work is very much simplified. He/she simply gives a rule and asks the pupils to verify it by application to several concrete examples. For example, students are told that the area of rectangle=LengthxBreath. Then a few sums are solved before the students. The students apply these formulae to solve these problems and they memorize it for future use. This method is very useful for small children because will small children we generally use story or telling method. This method glorifies memory, as the students have to memorize a considerable number of formulae and definitions. This method is adequate and advantages during practice and revision stage. contents back next

Strengths It makes the students think logically and scientifically. What is learned is retained longer and is better understood. The students are trained in problem solving and are able to solve similar problems later. Students go through a method of work both intellectually challenging and psychologically right. Weaknesses It does not fit all subject matter areas. It requires more time and is mentally taxing for slow learners. Lack of available resources for the proper performance of activities contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.12 Direction: Read the questions properly and write the answer in the given line. 1. What can you say about deductive method? ____________________________________________________________________________________________________________________ __________________________________________________________ 2. If you will use this method, how will you apply it? ______________________________________________________________________________________________________________________________________________________________________________ contents back next

Lesson 12 INDUCTIVE METHOD Objectives: At the end of the lesson the students are expected to: . differentiate deductive and inductive method; and . enumerate and explain the steps of inductive method. The Inductive Method is a method of solving a problem from the particular to the general. Through the inductive procedure, one way arrive at a fact, a principle, truth, or generalized. A lesson that utilizes the inductive method usually calls attention to distinct but related details that lead to the information of a conclusion, a definition, a rule, a principle, or a formula. The inductive method helps students discover important rules or truths for themselves through careful observation of among specific examples that will support the generalization, it makes meanings, explanations, and relationship of ideas clean to students and enable them to carry investigations for themselves independent of the teacher, through the master of the inductive procedure. contents back next

Example 1: Ask pupils to draw a number of triangles. Ask them to measure the three angles of each triangle and find their sum. They will find that the sum of the three angles of all triangles is 180: Example 2: Ask pupils to find the sum of two odd numbers like 3+5=8, 5+7=12, 9+11=20, etc. they will find that the sum of two odd numbers is an even number. Steps of the Inductive Method 1. Preparation – this step contain the: a. Appreciation b. Motivation; and c. Statement of the aim. contents back next

The perceptive basis may be review of old facts or lessons that may establish the proper mind set before the new lesson is introduced. The motivation sets the goal to achieve and gives direction to the activities to be accomplished. It is also in the motivation that interest is aroused. The statement of the aim is simply making the goal clear to the class, which may be in the form of a problem. 2. Presentation – specific cases or instances are presented to the class. There should be enough cases from which to draw a generalization. It is better to have more rather than too few cases. Otherwise, students may get into the habit of drawing conclusions from very few instances. 3. Comparison and Abstraction – it is in this step that the common element among the specific cases is delivered. Each case should be gone over as this is the step that prepares the student to state the generalization. Teachers sometimes hurry through this part, thinking that the students minds are as quick as theirs and have already perceived the common element. contents back next

4. Generalization – the common fact deduced from the specific instances is stated as a generalization, a rule, a definition, principle, or formula. The students ability to state the rule by themselves is the test of the success of the lesson. The success of the inductive method depends on clear organization of the lesson for which the teacher is responsible. 5. Application – this step test the student understands of the rule or generalization just developed. He should be able to apply it to other problems or exercises, if he understood it. For further mastery of the rule, it is also necessary to apply it. Merits of Inductive Method This method is psychological. The student feels interested in experiments, experiences and discoveries. This method fosters independence and self-confidence in the pupil which proves very useful in later life. contents back next

In this method, children discover the solution themselves. Hence it develops and encourages initiative and creative thinking. All that is learn using inductive method is remembered easily as it is self-acquired. In this method, the pupils observe and analyze particular objects of similar and different nature and try to arrive at general truth. Inductive method takes into consideration all the maxims of good teaching. The process of induction calls for perception, reasoning, judgments and generalization. Strengths It makes the students think logically and scientifically. What is learned is retained longer and is better understood. The students are trained in problem solving and are able to solve similar problems later. Students go through a method of work both intellectually challenging and psychologically right. contents back next

Weaknesses It does not fit all subject matter areas. It requires more time and is mentally taxing for slow learners. Lack of available resources for the proper performance of activities. contents back next

Name: __________________________ Date: ____________________________ Course/Yr/Sec; ___________________ Rating: __________________________ WORKSHEET NO.13 Direction: Read the questions properly and write the answer in the given line. 1. If you’re going to use the inductive method in teaching, what conditions must be present to ensure it work? ______________________________________________________________________________________________________________________________________________________________________________ 2. If you are going to choose a method, will it be deductive or inductive? Why? ___________________________________________________________________________________________________________________________________________________________________________ contents back next

Lesson 13 PROJECT METHOD Objectives: At the end of the lesson, the students are expected to: 1. understand and explain the project method; and 2. enumerate the guidelines of this method for effective use. This methods aims to bring practically designed experience into the classroom. Often conducted over a period of three to six months, the projects give students an opportunity to work in a team environment and apply theory learned in the classroom. There are some parts of the curriculum in which children are necessarily dependent on the teacher and others in which children can work more independently. Project work is more likely to continue the more informal part of the program, the part where they have greater autonomy in the development of their work than when involved in teacher directed instruction. contents back next

Project work can be seen as providing complementary learning opportunity to children in which they not only used to know how to use a skill but also when to use it. They need to learn to recognize for themselves the contexts in which the skill might be useful and the purposes which it can most appropriately serve. In project work they apply those skills in meaningful contexts. The project work can be seen as the part of the curriculum which is planned in negotiation with the children and which supports and extends the more formal and teacher directed instructional elements. Scope and Strategies This method is appropriate for any level, but is often employed for senior levels of education. Using projects usually requires a lot of preparation by the teacher. Some tips in this regard are as follows: Realize that the product of the project is not as important as the processes. contents back next

It is not important that the students determine an optional design. What is important, however, is that they experience the design process. It is important that the scope of the project is reasonable; care must be taken to ensure that the scope of the project is reasonable; care must be taken to ensure that the students are not overload. When possible, divide the project into section and set dates for the submission of each section. Assign project to teams of two or three students to allow in-depth efforts and to promote instruction among the students. Advantages 1. It is a teaching strategy that emphasizes “learning by doing.” 2. Constructing project develops the student’s manipulative skill. 3. The planned design of the project tests the student’s originality in choosing the materials to be used. They become resourceful and innovative. 4. It can be employed among students who are weak in oral communications. The finished product can serve as evidence of leaning achieved. contents back next

5. The completed project adds to one’s feeling of accomplishment and satisfaction, thus motivating students to continue constructing new projects in school and at home. 6. It instills the values of initiative industry and creativity. 7. Workings on a project in groups develop the spirit of cooperation and sharing of ideas. 8. In addition to learning a concept, student becomes productive and enterprising. Guidelines for the Effective Use 1. Assign the project to a student who is capable and interested. 2. The student must be clean about the objectives of the project including the criteria that finished product. 3. The design of the project must be carefully checked before the student starts. 4. The materials must be carefully selected as to suitability and durability to avoid wastage. 5. There should be minimal supervision as soon as the construction is going on. contents back next

Mathematics Project Ideas for High School Here is a selection of ideas for projects: 1. How is a Cricket Tournament Schedule worked out? How would you do such a schedule bearing in mind distances between locations of games, home team advantage, etc? Could you devise a good schedule for one of your local competitions? 2. How do major hospitals schedule the use of operating theatres? Are they doing it the best way possible so that the minimum number of operations is done each day? 3. Build a physical model to prove the Pythagorean Theorem. 4. Find as many triangles as you can with integer sides and a simple linear relation between the triangles. What about the special case when the triangle is right-angled? 5. Build a true scale model of the solar system. 6. Investigate the theory of pi and find the ways in which it can be approximated. 7. Construct a kaleidoscope. Investigate its history and the mathematics of symmetry. 8. Explore the history and use of the abacus. 9. Investigate card tricks and other magic based in mathematics. contents back next

Course/Yr/Sec; ___________________ Rating: __________________________ Name: __________________________ Date: ____________________________ WORKSHEET NO.14 Direction: Read the questions properly and write the answer in the given line. 1. Think a project in mathematics for high school? ___________________________________________________________________________________________________________________________________________________________________________ 2. Is it effective to use often times? Why? ________________________________________________________________________________________________________________________________________________________________________ contents back next

References Books Corpuz, Brenda, B. Salandanan, Gloria, G. “Principles of Teaching 1.” Quezon, City: Lorimar Publishing, inc. 2007 Ferer, Lourdes, M. “Teaching Mathematics in the Elementary Grades.” Hidalgo, Fe, S. “Tips on How to Teach Effectively,” Quezon City: REX PRINTING COMPANY, Inc. Highet, Gilbert. “ The New Book of Knowledge.” Scholastic Library Publishing, Inc. 2007 Salandanan, Gloria, G. “Methods of Teaching.” Lorimar Publishing Co. Inc. 2006 contents back next

Zulieta, Francisco, M. “ Principles and methods of Teaching.” 2006 URL http:www.google.com.ph/#hl=tl&source=hp&q=singapore+model+ method&btnG=Hanapin+sa+Google&meta=&aq=f&oq=sin gapore+model+method&fp=4eca90d4b93771a9 11-09-09 http://www.singaporemath.com/Mathemtics s/1.htm 11-09-09 http:/www.sleplessinkl.com/2009/04/24/relearning-maths-using- the-singapore-model-method 11-09-09 http:/www.singaporemath.com/Mathematics s/1.htm 11-28-09 http:/www.teach-kids-math-by-model-method.com/ 11-28-09 http:www/singaporemath.com/The Singapore Model Method for learning Mathematics p/smmlm.htm 12-26-09 http:/search.yahoo.com/search?p=singapore+model+method&togg le=1&cop=mss&ei=UTF-8&fr=yfp-t-701 10-25-09 contents back next

http:www.google.com.ph/#hl=tl&source=hp&q=singapore+model+met hod&meta=&aq+0&oq=singapore+mode+method+me&fp=ffec 8c89bd59b524 09-18-09 http:/home.sandiego.edu/~pmyers/singapore/kaur method of- model.pdf 11-15-09 Images 001.http:/images.search.yahoo.com/search?p=singapore+model+met hod&toggle=1&cop=mss&ei=UTF-8&fr=yfp-t-701 10-25-09 002.http:/images.search.yahoo.com/search/images?p=mathematics+l ogoo&ni=20&ei=utf8&fr=ytp-t-701&fr2=sq contents back next

Genalyn R. Obias graduated her tertiary at Laguna State Polytechnic University (Siniloan, Laguna) in the course Bachelor of Secondary Education (Mathematics as her specialization) in 2012. She had finished her secondary at Balian National High School (Balian, Pangil,Laguna) in 2008. And had finished her elementary at Cornelio Celis Dalena Elementary School (Pakil, Laguna) in 2004. contents back next

Marian Angeli A. Palma graduated her tertiary at Laguna State Polytechnic University (Siniloan, Laguna) in the course Bachelor of Secondary Education (Mathematics as her specialization) in 2012. She had finished her secondary at Balian National High School (Balian, Pangil,Laguna) in 2008. And had finished her elementary at Pangil Central Elementary (Pangil, Laguna) in 2004. contents back

Methods of teaching Mathematics

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Methods of teaching Mathematics