Introduction To Mathematics Education Hbmt1103.docx

HBMT1103

BACHELOR OF TEACHING ( PRIMARY EDUCATION WITH HONOURS)

SEMESTER SEPTEMBER 2018

HBMT 1103 INTRODUCTION TO MATHEMATICS EDUCATION/ PENGENALAN KEPADA PENDIDIKAN MATEMATIK

NO. MATRIKULASI

:

900116145695002

NO. KAD PENGNEALAN

:

900116-14-5695

NO. TELEFON

:

011-26267670

E-MEL

:

[email protected]

PUSAT PEMBELAJARAN

:

SRI RAMPAI LEARNING CENTRE

HBMT1103 Contents

1. Introduction

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2. Summary

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3. Discusion

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4. Suggestions

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5. Conclusion

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( 2-7 )

( 9-10 )

( 11 )

HBMT1103 Introduction Problem solving is a fundamental means of developing mathematical knowledge at any level. Problem solving gives students a context to help them make sense out of the mathematics they are learning. Problems can be used to introduce new concepts and extend previously learned knowledge. Although the definition of problem solving may differ to that of NCTM’s (1992), it, nevertheless, becomes the significant elements to be emphasized in the teaching and learning of mathematics. Teachers are expected to intentionally teach students on the heuristics of problem solving. Although teachers are free to chose the strategy suitable for his/her students, they are encouraged to follow those recommended by Polya (1974). Teachers are also encourage to simulate mathematical problems based on their daily experiences. More specifically, teachers are expected to provide varied experiences through students can work individually or in groups in tackling mathematical problems. The curriculum places heavy emphases on relationships between mathematics and real life problems. Problem solving in real contexts are considered essential in helping students appreciate mathematics. In short, problem solving becomes the focus in the curriculum. It is essential to understand problem-solving principles especially when it comes to dealing with real life problems. These problems can be very complex such as problems in economics, sciences, medicine and social sciences which can be translated into mathematical problems. Early exposure to problem-solving processes to young students will promote critical thinking at an early age. Introduction to different strategies in problem solving will give ideas and encourage them to acquire skills in solving mathematical problems, and later to apply the acquired skills to solve real life problems. In the era of technology and rapid expansion of knowledge, the importance of problem solving in the mathematics curriculum should be given more emphasis. Problem solving is the heart of mathematics. It does not only teach mathematical concepts, but also involves making decisions, thinking and doing reflection. Recently, problem solving has become the core of the mathematics curriculum in most developed or developing countries, such as Japan and Singapore. In fact, it becomes the major focus in most mathematics curriculum worldwide. Problem solving should be taught as early as students enter their primary school level. An early age exposure to problem solving will equip them with adequate knowledge and skills to solve problems including mathematical problems in mathematics classrooms.

HBMT1103 Summary : According to Smith (2001), problem-solving skills can only be developed after students have gone through different levels of problems. The levels of problem are illustrated in Table 1. In schools, students are exposed to different types and levels of problems. This enables students to develop different levels of problem- solving skills before they are able to solve non-routine problems. Problems can be posed according to type and level as follows: 1. Puzzles Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules as do multiplayer video games, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that satisfies the given conditions. Mathematical puzzles require mathematics to solve them. Logic puzzles are a common type of mathematical puzzle. The wagon must travel a path that is given by a mathematical function.Conway's Game of Life and fractals, as two examples, may also be considered mathematical puzzles even though the solver interacts with them only at the beginning by providing a set of initial conditions. After these conditions are set, the rules of the puzzle determine all subsequent changes and moves. Many of the puzzles are well known because they were discussed by Martin Gardner in his "Mathematical Games" column in Scientific American. Mathematical puzzles are sometimes used to motivate students in teaching elementary school math problem solving techniques.[1] Creating thinking (Thinking outside the box) often helps to find the solution. 2. Quizzes Mathematics Quiz is a small test administered to know the students’ knowledge or it is a short duration test used to know the student’s knowledge and understanding in the field of Mathematics.It is also one of the activities with the help of which a mathematics teacher can motivate and encourage the children to become truthful, active, and alert and to arose in him/her critical thinking importantly frankness straight forwardness. The mathematics quiz will go a long way in shaping and molding the personality of child towards mathematics. Mathematics quiz is a means to bring a path in the mind a way of habit of reasoning or thinking.

HBMT1103 3. Drill exercises A mathematical drill exercise is a routine application of mathematics to a stated challenge. Mathematics teachers assign drill exercises to develop the skills of their students. Early drill exercises deal with addition, subtraction, multiplication, and division of integers. Extensive courses of exercises in school extend such arithmetic to rational numbers. Various approaches to geometry have based exercises on relations of angles, segments, and triangles. The topic of trigonometry gains many of its exercises from the trigonometric identities. In college mathematics exercises often depend on functions of a real variable or application of theorems. The standard exercises of calculus involve finding derivatives and integrals of specified functions. Usually instructors prepare students with worked examples: the exercise is stated, then a model answer is provided. Often several worked examples are demonstrated before students are prepared to attempt exercises on their own. Some texts, such as those in Schaum's Outlines, focus on worked examples rather than theoretical treatment of a mathematical topic. 4. Simple translation A simple translation occurs when a shape is moved from one place to another. It is equivalent of picking up the shape and putting it down somewhere else. Vectors are used to describe translations. Translation is a term used in geometry to describe a function that moves an object a certain distance. The object is not altered in any other way. It is not rotated, reflected or resized. The rust-colored item is the pre-image, and the blue item is the image. In a translation, every point of the object must be moved in the same direction and for the same distance. When you are performing a translation, the initial object is called the pre-image, and the object after the translation is called the image. So, in the picture above, the rust-colored item is the preimage, and the blue item is the image. We know this because the arrow tells us the direction in which the image was moved. For other images, you might be told which image is the preimage, or you might be asked to find either the pre-image from the image, or vice versa. 5. Multiple-step translation In multiple-step one or more problems have to be solved in order to get the information needed to solve the question being asked. This lesson will provide help and guidance that will help solve these types of problems. Word problems are fun and challenging to solve because they represent actual situations that happen in our world. In any word problem, the true challenge is deciding which operation to use. In multi-step word problems, there may be two or more operations, and you must solve them in the correct order to be successful.

HBMT1103 Since word problems describe a real situation in detail, the question being asked can get lost in all the information, especially in a multi-step problem. Before rushing to solve the problem, it is worth your time to slow down and clarify your understanding. Be sure you know what is being asked, what you already know from the problem, and what you need to know in order to solve the question being asked. 6. Applied problems Applied math is a group of methods aimed for solution of problems in sciences, engineering, economics, or medicine. These methods are originated by Newton, Euler, Lagrange, Gauss and other giants. Modern areas of applied math include Mathematical physics, Mathematical biology, Control theory, Aero-space engineering, optimal planning, math finance. There is a fuzzy boundary between applied math and engineering and at the other side, between applied and pure mathematics. Applied math discovers new problems which could become subjects of pure math (like geodesics), or develop to become a new engineering discipline (like elasticity theory). Study of applied math requires expertise in many areas of mathematics and science, physical intuition and common sense, and collaboration skills. Applied math allows for many approaches to the problem, a choice of objectives, and variety of methods. 7. Routine problems From the curricular point of view, routine problem solving involves using at least one of the fourarithmetic operations and/or ratio to solve problems that are practical in nature. Routine problem solving concerns to a large degree the kind of problem solving that serves a socially useful function that has immediate and future payoff. Children typically do routine problem solving in primary level. They combine and separate things such as toys during their normal activities. Adults are regularly called upon to do simple and complex routine problem solving 8. Non-routine problems. Non-routine problem solving serves a different purpose than routine problem solving. While routine problem solving concerns solving problems that are useful for daily living (in the present or in the future), non-routine problem solving concerns that only indirectly. Nonroutine problem solving is mostly concerned with developing students’ mathematical reasoning power and fostering the understanding that mathematics is a creative endeavour. From the point of view of students, non-routine problem solving can be challenging and interesting. From the point of view of planning classroom instruction, teachers can use non-routine problem solving to introduce ideas (SET SCENCE stage of teaching); to deepen and extend understandings of algorithms, skills, and concepts (MAINTAIN stage of teaching); and to motivate and challenge

HBMT1103 students. There are other uses as well. Having students do non-routine problem solving can encourage the move from specific to general thinking; in other words, encourage the ability to think in more abstract ways. From the point of view of students growing to adulthood, that ability is becoming more important in today’s technological, complex, and demanding world. Level

Criteria

Own Word Problems

The problems require students to discuss or rephrase main ideas or procedures using their own words.

Level 1 Problems

These are mechanical and drill problems, and are directly related to examples in the book.

Level 2 Problems

These problems require an understanding of the concept and are closely related to examples in the book.

Level 3 Problems

These problems are the extension of the examples, but generally do not have corresponding examples in the book. These problems require problem-solving skills or original thinking and generally do not have direct examples in the book.

Problem Solving

Research Problems

These problems require Internet research or library work. Most are intended for individual research but a few are group research projects. Table 1: Levels of Problem

Strategies and techniques you can use to solve math problems. A useful starting point is a four step approach to math problem solving. There are four problem skills or steps need to teach in mathematical problem solving. Firstly is understanding the problem, As you carefully read the problem, trying to clearly understand the meaning of the problem and the question that you must answer, here are some techniques to help. Identify given information - Highlighting or underlining facts that are given helps to visualize what is known or given. Identify information asked for - Highlighting the unknowns in a different color helps to keep the known information visually separate from the unknowns to be determined. Ideally this will lead to a clear identification of the question to be answered. Look for keywords or clue words - One example of clue words is those that indicate what type of mathematical operation is needed, as follows: Then, Choose the Right Strategy .It step one has been done well, it should ease the job of choosing among the strategies presented here for approaching the problem solving step. Here are some of the many possible math problem solving strategies.

HBMT1103

Look for a pattern - This might be part of understanding the problem or it might be the first part of solving the problem. Make an organized list - This is another means of organizing the information as part of understanding it or beginning the solution. Make a table - In some cases the problem information may be more suitable for putting in a table rather than in a list. Try to remember if you've done a similar problem before - If you have done a similar problem before, try to use the same approach that worked in the past for the solution. Guess the answer - This may seem like a haphazard approach, but if you then check whether your guess was correct, and repeat as many times as necessary until you find the right answer, it works very well. Often information from checking on whether the answer was correct helps lead you to a good next guess. Work backwards - Sometimes making the calculations in the reverse order works better. Lastly, solving the problem and checking the solution If the first two steps have been done well, then the last two steps should be easy. If the selected problem solving strategy doesn't seem to work when you actually try it, go back to the list and try something else. Your check on the solution should show that you have actually answered the question that was asked in the problem, and to the extent possible, you should check on whether the answer makes common sense. Other than that, problem solving is a challenging for many students. However, many students never learnt the strategies of math problem solving. There are numbers of approaches for teaching problem solving in elementary mathematics. Four common problem solving strategies are, 1) Understand the problem Determined the problem what you need to find or which field of math concepts should be used to solve the problem. 2) Make a plan After you identify the problem select a plan to solve the problem. There are many strategies which you can use. Make a proper plan to solve any problem. 3) Solve problem in parts Geometry problem and many other problems include diagrams. Draw a neat diagram 4) Work backwards

HBMT1103 This is one of the most important strategies in math problem solving. Re-check the answer to the original equation. You can follow the backward method to re-check the problem.

These are the basic 4 ways to solve any math problem. But for solving some math problems you need to follow additional strategies. Sometimes just problem solving strategies doesn’t work you need to follow some strategies to build math confidence first then additional problem solving strategies. Such as : 5) Use step by step method. You can go online for step by step solution help. This helps you to work backwards. 6) Guess and check Follow guess and check method to solve small problems. But it doesn’t work for long problems. 7) Use logical reasoning Develop logical reasoning which helps you to solve math problem. Online math tutor, online games are the best way to master the logical reasoning. 8) Solve part by part Do not try to solve the whole problem at once. Solve part by part. This is one of the easiest ways to solve any problem. 9) Calculation Calculation is the main part of any math problem. Write down all the calculation part in one sheet. This helps you to re-check the problem easily. 10) Make problem simple Use math symbols and try to reduce the problem. Reduce word problems into equation. This makes your work simple and easy.

HBMT1103 Discusion of Importance And Challenges Of Teaching Problem Solving At Primary Level

Education is a challenging field where it changes according to current demands and needs. Educators are individuals who are responsible as trustees in this area. Mathematics is a living and growing thing. Mathematics is also an important foundation in the life of an individual Mathematic educators are advised to plan and implement an effective teaching and learning activities among students. The main goal to achieve is to develop the thinking, analytical, systematic, creative and critical thinking, skills in problem solving and ability to apply mathematical knowledge so that the individual can function in everyday life effectively and responsibly in appreciating the importance of mathematics. This is in line with the will of the National Education Philosophy in producing individuals who are balanced in terms of interpersonal, spiritual, emotional and physical. Problem solving is a very important aspect and is a key objective in mathematical learning. It is a very high learning form (Gagne, 1985). Students are expected to develop new knowledge and skills through problem solving. This problem arises in all mathematical topics including geometry. In solving problems in the form of problem solving cognitive skills students need to be well-developed so that they can utilize existing skills to solve mathematical problems. There are even some teachers who lack emphasis on math skills that involve problem solving. This skill is very difficult to learn if students can not master the concept in form and space. For weak students, they can not translate questions that are in the form of sentences or information to correct mathematical equations or inequities. There are also students who do not understand the question. This results in the use of the settlement rules or strategies that are not relevant to the question. The mastery of mathematical skills is important for students to demonstrate their own initiative to practice and solve problems. Primary school students are usually less interested in trying problem solving problems that are considered difficult. According to Polya (1957) problem solving is commonly associated with the use of mathematics in situations where the settlement procedure is not so obvious or significant.

HBMT1103 Suggestions : The use of resos in teaching and learning Mathematics is of an amount importance and increasingly gaining the attention of teachers and being the practice of primary school teachers. According to Piaget's theory focused on constructivism aspects, it is argued that children's thinking begins to manipulate objects to feel and develop Mathematical ideas starting from the pre-operational stage again. Generally speaking, Mathematics plays a very effective role in lesson of Mathematics starting from primary school again. Studies have shown that supporting the manipulation of auxiliary tool is very useful for mental development of pupils in connection with the Mathematical concept. According to Haylock and Cockburn (2003), he said that concrete experiences, pictures, languages and symbols can be important in understanding the concept of Mathematics for a student. As a result, the use of resume in teaching and learning Mathematics can involve students to perform activities. They are involved by manipulating concrete tools, symbols, languages and images which allow students to develop their thinking into abstract concepts. One of the fun games for pupils is that they can be in a situation controlled by their own learning. Pupils always try to solve problems and find solutions. Effective math games such as mental calculations in the minds of pupils. In addition, students are encouraged to discuss the Mathematical concepts in the game and the strategies used to help develop the language of Mathematics and give reason. In addition, the mathematical game designed is a match game that is often converted into a cooperative game where pupils support each other to achieve success. Math games can provide opportunities to develop skills related to Mathematical thinking ie predicting, justifying, and explaining. Mathematical games can also be used as homework or alone activities in the classroom. The most effective use of the game is when they are incorporated into a planned math curriculum. The next is related to high-tech Mathematics resumes is ICT. ICT is an information and communication band width technology that is available at school as a resume and tool to support teaching and learning of Mathematics. These include program robots, calculators, televisions, radios, videos, digital cameras, computers, software, internet access and interactive whiteboard. The use of ICT resumes can provide the most effective and effective way to achieve learning objectives. ICT is very useful in PdP Mathematics which provides a connection between Mathematical tasks using ICT software with activities that do not require the software. Examples of calculator usage which is an effective cognitive tool when they want to test their thinking in getting answers to the calculations made.

HBMT1103

Although there is little doubt that ICT provides a strong visual image and students are more motivated by the ICT software, for its unique use it is not necessary to increase the understanding of the Mathematical concept. Like the use of all Mathematical recipes, it is a choice of assignment, effective use of resos / software, the quality of teacher engagement and the opportunity given to the discussion is the basis of a successful PdP. However, this ICT resonance helps teachers especially when it comes to teaching that uses certain software. For example, the Geometric Sketchpad (GSP) software is widely used in the teaching of graphic, form and solid topics. Also, web pages or blogs with short notes are also displayed to students as one of PdP's resos.

HBMT1103 Conclusion In conclusion, problem solving in maths learning systems that emphasize the memorization and burden of training that the students do not understand make them drop out of the skills that need to be mastered. Hence it makes their interest in learning to continue blurring. Students often make assumptions when they try to understand something the teacher is trying to convey and that assumption is wrong. This means that the development of concepts in their minds is different from what the teacher wants to achieve. An effective teaching and learning session should contain elements such as exposition or presentation of lesson content by teachers, discussions between students and teachers as well as the students themselves. In addition, Problem solving learning should also have appropriate work practices, drill training for basic skills, activities problem solving and research activities of various ways of solving problems. Therefore, the right pedagogy is to realize the various problems that may arise and the creativity of teachers to create meaningful and comprehensive teaching and learning. Teachers need to have strategic planning to determine how appropriate methods, techniques, approaches to teaching each skill contained in the measure. Issues or problems faced need to be addressed to the best of strengthening the human capital of the country with the complete knowledge of science and mathematics which is the catalyst for the development of a country