You are what you are and where you are because of what has gone into your mind. You can change what you are and where you are by changing what goes into your mind.
This week marked as the 6th week, two weeks before we closed the month of February 2009. Time move so fast, sometimes I feel that I cannot keep up with its pace, the works doubled up and a lot of challenges triggered. I strongly believe that, a human’s life is not complete without challenges and problems, only one who can endure until the end is the winner. I always shout at myself that no pain, no gain every time problems arise. This week is kind tough for me especially in thinking what will be the issues will arise when I officially became a teacher within two years from now on! Can I become a great teacher like Madam Lam Saw Yin? Madam Lim Bee Leng? Mr. Koo Teck Hock? Madam Juliana Osong? Can I control the class very well or can I teach mathematics in the most effective ways? I feel headache just to concern what will be happen in the future. Based from reading from Article X, different issues arise in different countries. For example, in Malaysia, teaching Maths in English is very challenging especially when we as the new teachers transferred to a place where its population cannot understand the English at all. If they don’t understand any words, how can they understand the concept mathematics? As we concern, the problems which may arise here is the language factor. Other issue which maybe existed is the method that we as the teachers will be using whether based from the text and workbook only, teaching kits or both. So at the end of the week, a debate between groups from Maths 1 and Maths are being held. From the debate, a lot of issues were highlighted and recommendations were being discussed. Another interesting debate is about; Training a mathematics generalist or a specialist in handling primary mathematics in school: Its impact on children’s learning of mathematics. This sure makes the whole class on fire. For me, I prefer the generalist especially for those schools who lacks of teacher. Flexibility and sophisticated is the key of the generalist. He maybe can be English and Mathematics teacher; two in one, more effective and efficient.
Reflection Week 4 This week I have learned about algorithms. It is about how to solve a math problem by using mental calculation. At school we need to make sure the students understand how to solve a problem and also to make sure they know why we use certain steps to get the answers. I believe that when the students understand why they use their problem solving steps to solve a math problem, they would be able to answer as many questions in a various way of solving it. I also have learned a few new addition steps which called Partial sums method and column addition method. It is much faster to calculate mathematical problem using this two addition method than the standard method we usually use. Other than that, we also learned a new subtraction method for me that can easily and much faster to solve a mathematical problem. During this week also, we have done a little bit of discussion whether calculator is suitable to be used at primary school these day or not. For me, I prefer using calculator at school. This is because students can learned number much faster this way in fact children like to play with something when they study so using a calculator might attract them to be involve in learning session.
In the 5th week, we came to our last part of the operation sense and computations. Algorithms known as computational skill with paper-and-pencil procedures have been viewed as an essential component of children’s mathematical education. In mathematics, computing, linguistics and related subjects, an algorithm is a sequence of finite instructions, often used for calculation and data processing. It is formally a type of effective method in which a list of well-defined instructions for completing a task will, when given an initial state, proceed through a well-defined series of successive states, eventually terminating in an end-state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as probabilistic algorithms, incorporate randomness. Beside that, each of us needs to summarize and present our view on whether or not children in primarily school used calculators and the way in which they could be used. This open discussion sparks many different opinions; some oppose and many support the idea. For me, appropriate calculators should be available to all students at all times. My point is as students’ mathematical knowledge grows and changes, so does their need for calculators. Children outgrow calculators just as they outgrow shoes. The calculators needs of students in primary grades are different from those of students in secondary and college school. A calculator is a device for performing mathematical calculations, distinguished from a computer by having a limited problem solving ability and an interface optimized for interactive calculation rather than programming. Calculators can be hardware or software, and mechanical or electronic, and are often built into devices such as PDAs or mobile phones. In most countries, students use calculators for schoolwork. There was some initial resistance to the idea out of fear that basic arithmetic skills would suffer. There remains disagreement about the importance of the ability to perform calculations "in the head", with some curricula restricting calculator use until a certain level of proficiency has been obtained, while others concentrate more on teaching estimation techniques and problem-solving. Research suggests that inadequate guidance in the use of calculating tools can restrict the kind of mathematical thinking that students engage in. Others have argued that calculator use can even cause core mathematical skills to atrophy, or that such use can prevent understanding of advanced algebraic concepts. There are other concerns - for example, that a pupil could use the calculator in the wrong fashion but believe the answer because that was the result given. Teachers try to combat this by encouraging the student to make an estimate of the result manually and ensuring it roughly agrees with the calculated result. Also, it is possible for a child to type in −1 × −1 and obtains the correct answer '1' without realizing the principle involved. In this sense, the calculator becomes a crutch rather than a learning tool, and it can slow down students in exam conditions as they check even the most trivial result on a calculator.
For this week, we focused on the role of mental computation and computational estimation. Mental computation is computation done all in the head that is without tools such as a calculator or pencil and paper. In fact, research has documented a wide variety of mental computation techniques that children have created on their own and that make sense to them (Fuson, 2003). During tutorial hour, we implement the activity based from Groves which is known as ‘How did you do it?’ ,in this activity, our lecturer, Madam Lam, present an additional question which need to be done using mentally. The responses were variety; some using the mental strategy by adding from the left, counting on, making tens, doubling, and making compatibles. For me, it’s hard to explain. I got the answer by automatically and it’s so mystery. When I look up for this type of mental computation in the internet, it’s been shown that it’s very rare to find this type mental strategy. It’s like when someone hit your knee, your knee will automatically reflects. This phenomena known as automatic reflects. Well I suppose that can be best describing my unique mental strategy. Beside from learning the mental computation, we also learn the estimation skills. In my opinion, estimation is very helpful especially when we need to estimating a very large quantity. For example, we need estimating criteria when we doing research to estimating the population of tourists who visiting Sabah every year or estimating the number of bacteria which living in the microorganism habit when doing the microbiologist area. Learning about estimation gives student their first encounter with an area of mathematics that does not focus on exact answers and yet is natural part of mathematics. Estimation involves a different mindset from the mindset that says only an exact answer will do.
This week I have learned about the number operation and basic facts. Besides that we also have learned week 4 topic that is operations sense and computations. In number operation and basic facts, we have learned about how to develop meaning involving the operation addition, subtraction, multiplication and division. What we need to do is to illustrates the children using model and concrete things which to make them understand how to add, subtract, multiply and divide numbers.
We have learned the basic facts of addition which involved two, one-digit addends and their sum. The basic subtraction facts rely on the inverse relationship of addition and subtraction for their definition. Basic multiplication facts each involve two, one-digit factors and their product. While basic division facts rely on the inverse relationship of multiplication and division. We can see the relationship between each operation by these facts.
In operations sense and computation, it explains about mental computations and computational estimations. In this topic, we are given a mathematical problem and need to solve it using mental computations. After that we need to explain how we do it or what method do we use to solve it. The purpose of this mental calculation is to make the children to be able to find their perfect strategies and method to solve the problem. After that they need to reflect back what they have used to solve the problem.
This week tutorial task is to prepare an activity which can be used at school to make student learn mathematic while having fun. My group prepares an activity that called addition BINGO. The teacher draw a card which contains mathematical problem and the student as participant solve the question using mental computations then they tick the answer on the paper given. Whoever finishes ticking five numbers in a row or column will win the game.
This week is before the Chinese New Year's holiday break. It does not mean we can be relaxed but we have to put extra concentration as our syllabus getting harder and harder. Our target topic for this week is Number Operations and Basic Facts. In this chapter, we learned on how to develop meanings for the operations of addition, subtraction, multiplication and division.
It is very important that students see mathematics, and the calculations they perform, as part of their daily life. Providing opportunities to apply basic concepts and operations in daily activities will reinforce students' skills and motivate them to progress in mathematics. They can use addition to figure total amounts of toys or snacks, and to keep track of their bank accounts or team equipment. Students can use subtraction to make comparisons between what they have and what they need for a game or other activity, to budget, and to calculate remaining items as they are used, or to calculate change when a purchase is made. They can multiply to figure larger totals, and to transform units from one measure into another. They can divide to determine equal portions of items, or to figure daily averages for sports scores or percent scores for quizzes or games.
In order for students to calculate using these four basic operations, they must first have developed basic concepts (including more, less, many, etc.), one to one correspondence, the concept of sets, and basic number sense. As students begin to learn to calculate, the following teaching considerations should help:
Emphasize concept development rather than process or rote memorization.
Apply operations to real life situations which are of interest to the student (e.g., provide opportunities for students to determine quantities of materials needed to play a game or complete a project and to estimate the price to purchase these materials). At first, provide examples for the student, then ask the student to provide his or her own examples which he or she sees as relevant uses of different operations.
When students are using manipulatives, encourage them to search the entire "field" to make sure they are aware of all the objects with which they must work. Using trays or mats can help to identify this field and the area they must search.
Word problems are very effective since they involve practical application of skills. To assist students in developing the skills necessary to solve word problems, it may be helpful to provide a problem solving model. First, identify the specific kinds of information needed in a particular problem; then provide two or three choices of operation statements to solve the problem. Eventually, students will be able to identify appropriate operations independently.
Teach the concept of complements or partners for addition, subtraction, multiplication and division. For example, the number 5 is made up of 2 and 3, 1 and 4; 24 is made up of the factors 8 and 3, or 2 and 12, etc. This concept not only increases the student's ease with number facts; it also facilitates mental mathematics.
When teaching facts, focus on 2 or 3 related facts at a time. Emphasize accuracy first, then speed. Maintain a chart of mastered facts to help the student recognize progress.
Small flip charts can be provided at a student's desk, with cues for steps in particular types of problems as a reference or reminder.
During the tutorial, our class were divided into four groups and each group required to prepare one activity using cards and exchange the activity with the other groups so that they can play all the 4 activities. Our group did a activity which was called Addition Bingo. The leader draws a card and reads the addends on it. Each player covers the sum on his or her Bingo card. Not all sums are given on each card. Some sums are given more than once on a Bingo cards, but a player may cover only one answer for each pair of addends. The winner is the first person with 5 markers in a row.
This week, we had learned on Number Operations and Basic Facts and also week 4 lecture topics which were Operation Sense and Computations. We do have our replacement class on Tuesday due to the Chinese New Year holidays. For number operations and basic facts, we learned on how to develop meanings for the operations of addition, subtraction, multiplication and division. Madam Lam teaches us on how to illustrate the children using models or concrete things in order to show the children how to add, minus, multiply and divide numbers.
Besides, we also had been exposed to the basic facts of addition which is involved two one-digit addends and their sum. The basic subtraction facts rely on the inverse relationship of addition and subtraction for their definition. Basic multiplication facts each involve two one-digit factors and their product. While basic division facts rely on the inverse relationship of multiplication and division.
Operation sense and computations involved the role of mental computation and computational estimation. This topic required us to calculate the numbers mentally and try to explain how we did it. The focus for these mental calculation activity is children find their own way and strategies and reflecting on what they are doing.
For our tutorial, we were divided into four groups and need to prepare one activity using cards and exchange the activity with the other groups so that they can play all the 4 activities. My group did a activity which was called addition Bingo. The leader draws a card and reads the addends on it. Each player covers the sum on his or her Bingo card. Not all sums are given on each card. Some sums are given more than once on a Bingo cards, but a player may cover only one answer for each pair of addends. The winner is the first person with 5 markers in a row.
For the next tutorial, we were divided into groups and complete the activity 1 and make a comment on the findings regarding to the activity.
This week, we had learned on the numbers’ place value. We were focusing on how to teach primary pupils to understand what actually the place value of numbers. Personally, I realized that it was not easy to teach this topic because there were certain terms which were difficult to convert into words. For example, when we pronounce 432, we won’t say four hundreds three tens and two ones. At first, I couldn’t explain why it was but after having a discussion with the whole class including my lecturer, I realize that the place value and the numbers are different.
Besides learning that, we also had been asked to make number expanders which were related to place value topic and also making a hundred numbers chart. For me, that was an interesting experience because we had never been it out before. Our group have quite numbers of ideas during making the expanders and we manage to complete three number expanders.
For our tutorial, we need to carry out some ideas on how to show 201 and 120 to the pupils. Other than that, we also need to examine a primary mathematics text book and find as many as place-value model had been illustrated in the book and then comment if the text book provides a good choice of models.
Week 2 (12/1-16/1) This week we further our study in the chapter of the whole numbers. This week we focus on the subtopic of grouping or trading, developing place value and regrouping and renaming. This subtopic showed to me how, for example, 201 and 120 would be represented with three different place- valued models. In order to learn more about on how to teaching the place value, me and my group analyse place-value models that are illustrated in the text books. From our analysis, I found that there two types of models which is the proportional models and the non-proportional models. In proportional models the material for 10 is 10 times the size of the material for 1; 100 is 10 times the size of 10, and so on. Measurement provides another proportional model where children use interlocking cubes to measure their height and then form groups of 10. Meter sticks and centimetre cubes could also be used to the model besides base ten blocks, beans glued in groups of 10 on a stick and tongue depressors bundled together. Non proportional models do not maintain any size relationships. Money is a real world example of non proportional model where size relationships are not maintained. For example, ten of 20 cents are bigger than two 50 cents but are a fair trade in our monetary system and yet they are not proportional in size. From our group activities, we managed to build our own expanders. This model connect the base ten blocks to number symbols in an expanded form and can be used effectively to review key ideas. For example, while 123 is composed of 1 hundred, 2 tens, and 3 ones, if we dividing the number by 3, it would be easier to think of 123 as 12 tens and 3 ones. These encouraging children to name the same number in different ways promotes number sense. Then our next group activity, we create our hundred charts which provide many opportunities for counting and doing mental computation. For example, 43+30 can be determined by counting mentally, 43, 53, 63, and 73 and is a natural by product of counting by tens on the hundred charts. From theses activities, I feel there are a lot of room for improvement that I should make in order to ensure my teaching aids can performed better and efficient in educating the pupils.
This week remarks the new beginning of my third year in completing my degree under the program known as PISMP of Mathematics. To begin with I would like to thanks to all my lecturers especially Madam Lam and my friends who had contributed a lot of supports and knowledge in order to help me in my study for the past 2 years. After a brief of introduction of our new subject, I begin to knowing this course. Our course is Teaching of Numbers, Fractions, Decimals and Percentages.
In this whole week, I learn about the definition and the function of the number sense is all about. From my reading, I realised that number sense is not a finite entity that a student either has or does not have. Its development is a lifelong process, and in early childhood and elementary school number sense development involves several stages; Pre number development; Early number development and Number development. Besides my reading from Helping children learn mathematics by Robert E. Reys, Madam Lam also suggested to me to read an article entitled ‘Two needed revolutions’ by A.J McIntosh from Challenging Children to Think When They Compute which I found really interesting and I am feel amazed with the uniqueness of number.
At the end of the week, we had some great time where we were divided into several group to play some game involving number sense. Suk Ling and me designed a game which identical with the game of ‘Snake and Ladder” but we did some modified so that it is suitable to primary students and focus in building their number senses. Overall, this week is so great and I am so enthusiastic to be continuing my study.
This week we have learned about the number place value which we need to know how to teach and make the student understand what place value really is. From what I have learned during the class, we can use many learning kit to help us make the student understand about place value. That including using a card to show them which place value the number belongs to. This is including how to pronounce the number in words. For example 689 which in words we pronounce six hundreds and eighty nine. In pronunciation of numbers, we don’t have to mention the numbers place value like tens and ones.
For our group work, we have made a number expander which first it shows the number then when expended we can see the numbers place value. this expander can be use effectively while teaching at school but to make it easier to use we should make it bigger and build a flat form for it to so that the student can see clearly.
After that, we have been given a tutorial task which is to think of a way to show 201 and 120 to the students. We also need to find as many place value model we can find then give any comment to it whether it is suitable or not to be used in teaching.
This week was the first week we attend the class in the year 2009. This semester Madam Lam will teach us on Teaching of Numbers, Fractions, Decimals and Percentages. We had learned on number sense using the suggested book. Now I realize that the book was really useful especially when completing my practicum later. Number sense is about to understand the number concepts and the operations on these numbers. Therefore, people with number sense are able to use the numbers effectively in daily life.
Beside that, we also learned on counting principles. There were 4 important principles that we need to know and understand which were one-to-one counting, stable order rule, order irrelevance rule and cardinality rule. Each one of the principles has their own terms.
For our tutorial task, we had been asked to create a mini game for micro teaching which was related to numbers. I and delver were prepared a game using picture. We cut the picture into pieces and label it with number. So, the pupils with rearrange the picture based on the numbers. Therefore, they need to know the numbers sequence first.
This week I have learned about number sense. Number sense can be described as an understanding of number concepts and operation on these numbers, the development of useful strategies for handling numbers and operations, the facility to compute accurately and efficiently, to detect errors, and to recognize result as reasonable. In my understanding, people who have number sense are able to understand numbers and can use them effectively in daily life situations.
In children level of understanding, their number sense is more on using arithmetic computation for example 47 + 53. Most of children will use a written calculation to solve this kind of question but for those who have already achieve number sense they can straight away solve this problem using their mental Arithmetic skill. But of course number sense is not easy for all children to get because it will develop slowly through life.
This week also we have done and create some mini games for micro teaching. I couldn’t join this activity because I’m not feeling very well that day. So I just join Zool’s group. The game that they create is a picture maze that needs to be arranging accordingly. To get the answer, just arrange the picture according to the number from number one to ten.
