## Friday, 2 September 2011

### Reflection on Session Six (31/8/2011)

Session Six (31/8/2011)
It was the last session for this Elementary Mathematics module. We were told not to use the word “problem sum” (word that was taught to me by my teachers years ago and in turn has been used by me rather frequently) when teaching children. Instead, word such as “story problem” or “word problem” should be used as they are more appropriate.
Quite a number of things were covered and we had two interesting activities which were (1) counting the distance from the upper landing of the staircase to the bottom and (2) doing of cube. We were told to bring our ruler along to the Circle Line Station and best part was quite a number of us were without ruler – and one of them was me (how embarrassing)! Nevertheless by working with the rest of my group mates, we still managed to find the distance. The steps that we took to do the calculations are as follows:
·         First we counted how many steps were there in each flight of stairs.
·         In total there were 62 steps from top to bottom and each step measures about 15cm.
·         Therefore, 62 X 15 = 930cm, which we then converted to meters (930/100 = 9.3m)
As I was doing this reflection, it occurred to me that we did not actually count the space in between the steps and as far as I am concerned it should be included as the spaces formed part of the steps leading to the bottom. Therefore, if my deduction is right, the above answer may be wrong. Unfortunately, class was over and we could not possibly go back to the station to do another measurement.
Another activity that we did was to make cube to fit in 15 kidney beans. We underestimated the volume of the cube so much so that our cube was far too big. It could probably fit hundreds of beans!
A few others important lessons that I took with me are the Assessment of Children and Incidental Teaching of Time. Careful consideration must be given to ensure the validity of the assessment. The objectives of the lesson must match the instrument used and the assessment must be reliable regardless of other factors. As for teaching of time, all along it has been our practice to make children “learn” time by having proper lessons and by “drawing” the clock face. Only after Dr Yeap explanation was I able to see the practicality of teaching time in an
incidental ways. Connecting time to events would be one of the meaningful ways to teach children.
In conclusion, the whole module has been an “eye-opener” for me in relation to teaching mathematics to young children and I sincerely hoped I would be able to make a difference in the lives of the young children when they learn mathematics.
A Big Thank You to You Dr Yeap!

## Tuesday, 30 August 2011

### Reflection on Session Five (26/8/2011)

Session Five (26/8/2011)
It was the fifth session with Dr Yeap. More and more complex problems were discussed. We continued with fraction. Though Dr Yeap tried his best to explain and provided visualization, I still found it rather tricky and a great challenge to try comprehending the lesson. However, after a few attempts, yes, I managed to do it. Being not mathematically inclined and the way I was taught mathematics when I was in school could be the contributing factors as to why I had problem viewing and understanding the problem sums well. Hence, I am determined to make the children learn mathematics in the proper ways it should – the CPA Approach.
We also learnt about Bloom’s Taxonomy.  I looked up the meaning of the word in thesaurus and a few other words came out such as classification, categorization, arrangement and such. Out of the six levels in Bloom’s Taxonomy, three levels were used in mathematics and they are knowledge, comprehension and application levels.
We also did graph and was told to chart the number of pegs we received as tokens for our class participation. I saw quite a number of graphs being done. For my group, we did the line graph. In total till session five, I collected seven pegs. It was encouraging to receive that number but I actually hoped that I could participate more if only I understand the problem sums being discussed better and able to “see” and provide reasoning in the mathematical ways.
Dr Yeap did mention that in Singapore children have the opportunity to learn the same topic again and again albeit at a higher level (The Spiral Curriculum). My question is – “what happens if a child is not able to master the very basic level of a particular topic and he is exposed to the same topic but at higher level?
Well, we have another lesson for mathematics. As of now, I must say that I have learnt quite a number of new things pertaining to mathematics and am still looking forward to learn more…

## Monday, 29 August 2011

### Reflection on Session Four (25/8/2011)

Session Four (25/8/2011)
It was very interesting when we played the number guessing game. I was amazed when Dr Yeap managed to guess the number correctly. He’s a real “magician”! We were taught the logic behind the trick. After trying for few times and understanding the ways it works, even I could do it (by either using multiple of nine or algebra). Back in my centre, I tried this game with some of my teachers and I got the answers right.
We moved on to fraction. Using the colored paper we tried different numbers of fraction. From the various hands on experiences, I know realized clearly that being equal does not have to be identical. We have to look at the area of each shape to justify whether they are of equal size. Throughout the various activities, I tried my best to contextualize the ideas to Early Childhood setting.
On usual occasion, my teacher will teach children whole and part by showing them pictures on the board before letting them to do it in the worksheet. Hence, I adopt what I have learnt in class, brought forward and simplified the activity (finding equal parts) with my K2 children. They were able to find the equal parts of squares, rectangles and triangles. But I did not go beyond asking them to count the quantities of the shapes.  While I enjoyed each and every session, my main objective is that I should be able to put into practice what I have learnt and most importantly, it must be beneficial to the children. And I am glad I managed to do just that.
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 Children having the hands on experience on finding equal parts.
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 Surprisingly, they did not stop at half of equal size but moved on to quarters as well.

## Sunday, 28 August 2011

### Reflection on Session Three (24/8/2011)

It was a continuation of yet another interesting session. It started off with Peggy asking us to try guessing the meaning of lesson study. She threw the ball to me and I had to guess what lesson study was. I have not heard of lesson study before but from the word I guessed it had something to do with teacher conducting the lesson, look into the areas that can be improved, comes out with plan of action and finally conduct an improvised lesson. Many other suggestions were given pertaining to lesson study such as it involved curriculum, the lessons, the observations and such. Amidst all these suggestions, one of the cores (I would say) would be the incorporation of range of challenges depending on children’s learning abilities in all the lessons.
From the two videos that we watched, I have learnt a great deal on the factors that categorizes good teaching of Mathematics. Most often, the categories are lesson content itself, teacher’s attitude, sitting arrangement, classroom management and communication, be it teacher-child or child-child interaction.
Indeed it was very practical to conduct lesson study as there is a direct involvement with children; hence we will be able to see the “real” result of our lesson. This topic on lesson study, somehow reminded me of the module that I did before on Action Research. To me, it seems somewhat similar. However upon clarifying with Peggy, I than realized the difference – Action Research is dealing with children in the controlled group whereas lesson study is just as a whole class. As this session focused more on observation and teaching strategies and “less” on numbers, I understood it better and was able to participate more actively – something which I very much wanted to do in every class session.
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We also did the manipulative and tried to construct various structures bearing in mind that one should not be similar to the other (when it was rotated or in different orientation). From this activity and many others, I realized that great focus is given on being equal but not identical. Till now, I am still trying to see the rationale behind it apart from seeing it mathematically.  Could it be that – As teachers, we must be flexible in accepting various ideas and suggestions from the children when they are attempting to solve problem. Also, it could be for the purpose of stretching children’s logical and creative thinking skills. (I am still pondering on it).
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 ﻿﻿Doing of addition activity by first using concrete materials. Bearing in mind the different level of children's abilities (as always mention in class) I gave each child different amount of ice-cream sticks to work with. For the average and lower ability level children, they were given lesser ice-cream sticks such that the total amount will either be equal or fewer than 9 and the total values will be equal or fewer than 5 respectively.
Those with higher ability level were given more ice-cream sticks so that they can do addition with total values up to two digits number.

## Friday, 26 August 2011

### Reflection on Session Two (23/8/2011)

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We started off by doing the quiz on cardinal number. By virtue that the name was “quiz” I was already shaken. With the time given (15 minutes) it was almost as good as I was not able to do it at all! As mentioned in my first session reflection, due to my lacking of mathematical skill or rather my phobia of it, I need a little longer time to think it through every maths problem. I can’t imagine that I am actually doing maths quizzes at this period of my learning journey again. Nevertheless, despite my shortcoming, I am not going to give up. Rather, I will do my best to learn so that I will be able to use the knowledge gained with the children in my centre.
﻿﻿We learnt new word today – subitize. Once again, I tried to put into use the concept behind it. Yes, it works again! Most of the K2 children were able to subitize number from one to seven. There was also a group of K2 children with higher abilities who were able to subitize numbers up to ten (I had the opportunity to really see children with differentiated learning abilities).While this game was going on, at the back of my mind I was recalling the fact about ‘differentiated learning’ – that was emphasized repeatedly by Dr Yeap and Peggy). Now that I know the importance of it, I will work closely with my teachers to ensure that we will take into consideration the differentiated learning aspect every time the teachers conduct lesson.
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Figure 3: Children were so excited to participate. As they were told to voice out the number of circles without counting them individually, some of them covered their faces – just to make sure they did not ‘accidentally’ do the counting
Figure 4: Some of the higher ability children were able to subitize the number of circles correctly while some others actually need to count them.
After going through the session with the children and told them about subitizing, I tried my best to then explain to my teachers about the concept and how they could detect whether the children are doing it or not. While in the process of explaining, one of the children joined in our conversation and said to her form teacher, Teacher Zubaidah, we can write this word in our “Word Bank book right?” I was thrilled to think that this child was able to extend her learning from maths to language. With that, the rest of the children joined in chorus and agreed with the idea. This evident really proved to me that children do not learn in isolation. They are able to extend their learning into other subject areas. Hence, it further affirmed my understanding that every learning opportunity can encompass the different domains of development.
Moving on with the stick game, we learn about the good and bad numbers. I managed to detect a few patterns in relation to the game but have yet to try playing the game with the kindergarten children in my centre. And in everything we did, justification is necessary so as to ensure we understand the logic behind it.
Many other concepts were taught. In short, children learn in the following ways:
v  learn procedure (procedural understanding)
v  learn conceptual (relational understanding)
v  learn convention (conventional understanding)
And another important understanding – Mathematics is a tool (vehicle) for learning and thinking for the development and improvement of child’s intellectual competence.
Looking forward to Session Three...

### Reflection on Session One (22/8/2011)

A 1000 summers”, we started off with counting the position (cardinal number in space), trying to find the position of the letter “N” in the name “BANHAR”. As I am not very mathematically inclined, it was initially rather daunting having to attend sessions in mathematics. While I was still trying to internalize the concept, another concept emerged. More and more mathematical jargons were introduced – ordinal, cardinal, number in space, growing pattern – to name a few. In a nutshell, we had gone through the five process standards that are, problem solving, representation of ideas, justification of the way we used to solve problem and such. These processes were done through various activities such as poker, the cookies, number sense and ten frames.
Honestly, even with the space ordinal for my name, I had problem initially doing it (so much so that I did it in the primitive was – as Dr Yeap put it) until my partner, Rahimah highlighted the method to me.
﻿﻿ Figure 1: Finding the position in space for the letter “D” using the primitive way until after awhile than I realized there was a pattern to it – that is, the multiple of 8

POKER GAME
I tried to do it with the children in my centre. I did the bigger version of the so called poker card so that it will be easier for the children to seen the dots on the cards. The purpose was two folds – to do reinforcement on spelling of number words and for me to practice my “skill” of doing it. IT WORKS! Children were very engaged and best of all – they thought it was magic! I feel this method of learning is really effective as children enjoyed the session while at the same time undergoing a learning process without sheer pressure.

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 Figure 2: Children were so excited when then number that they spelt appeared.
This was the first session, I must say that I enjoyed the session and realized that maths can be taught in a fun way. Nevertheless, I could not help feeling worried for the fact that will I be able to participate as actively as I wanted when I actually have to take a longer time trying to understand each concept taught. Many a times during the session, I was just “lost” and did not know how to be involved in the discussions as I need longer time to try to understand and make meaning the concepts and activities conducted.
Well….moving on, I look forward to the next session and hopefully I will be better “enlighten” as the session goes by...
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## Thursday, 18 August 2011

### EDU 330 Elementary Mathematics - Pre-course reading, Chapters 1 & 2

Reading Reflection – Chapter One
Teaching Mathematics in the Era of the NCTM Standards

v  Important tools for teacher in order to teach Math efficiently - knowledge of mathematics and how student learn mathematics.
v  Teaching mathematics should not only focus on doing exercises and testing, rather it should be more holistic - focussing on mathematical thinking, reasoning and problem solving skill.
v  A each level, children's learning should be on certain focus, go in more depth and there should be connections.

My Reflection - from my understanding, I feel that children's learning of mathematics should be in the form of spiral curriculum. That is, at each grade level, teacher should build upon children's knowledge on mathematical concepts from their previous experience. Hence, it will help to form concrete understanding on the various concepts taught.

Principles and Standards for School Mathematics

The Six Principles
1. The Equity Principle
Regardless of personality traits, background or physical challenges, all children should be given the equal opportunity and guidance to learn mathematics. This support or guidance should be equipped with high expectation for all the children.

My Reflection - From my understanding, to have high expectation for all the children and to support, sound similar to Vygotsky's theory of Zone of Proximal Development. Meaning to say, teachers should set high expectation for the children to acquire as much knowledge as possible through scaffolding.

2.      The Curriculum Principle

Teachers should enforce the understanding that mathematics learning is as a whole. In other word - through the various mathematical skills, it can be integrated into other forms of learning from other disciplines.

3.      The Teaching Principle

For effective teaching, teachers must focus on the “KWL” – what children know, what children need to know and finally how they have learned it.

4.      The Learning Principle

Learning with understanding. Building upon new knowledge based on previous knowledge gained – spiral curriculum.

5.      The Assessment Principle

Using assessment as the mean to gauge children’s level of understanding. It also serves as a mean for teachers to plan future lessons and ways to further enhance children’s learning.

6.      The Technology Principle

In this IT savvy Era, children could be exposed to mathematics via the use of technology. For instance, by using computers, children could learn the different mathematical skills.

The Five Process Standards

The Five Standards are inter-related and should be viewed as a whole. Problem solving is essential in mathematics and children must be able to provide logical reasoning for their answers. They should also be able to communicate and explain their ideas and make connection to past experiences, real situation and linking to other areas. Finally, children should be able to use symbols, chart, graph and such to express their ideas.

Six Major Shift in the Classroom Environment
These shifts are necessary in order to allow children to develop mathematical understanding.

The Teaching Standards
Of the seven teaching standards, I would focus more on three of the standards, namely,
v  Standard 4 – Learning Environment
To me, learning environment plays a vital role in ensuring that children are able to maximize their learning potential based on that particular subject. With the appropriate materials, space, time and such, children will be able explore the concepts that they are learning in depth.
v  Standard 6 – Reflection on Student Learning
Reflection and follow-up with plans of actions are something that teachers should do without. Hence, I make it a point for all my teachers to do daily reflections on children learning. From these reflections, teachers will have better information pertaining to every child’s learning performance before deciding on the next plan of action.
v  Standard 7 – Reflection on Teaching Practice
Besides reflecting on children learning, teachers also must reflect on their teaching style and the content covered. It is essential to do so for the purpose of improving their teaching method and also to gain more insights about their teaching.

My Reflection
In a nutshell, I have gained better understanding in teaching mathematics. I totally agreed that the teaching process should not only focus on practices and assessment, rather, it should be more of in depth learning of the various concepts – dealing with concrete materials and real-life experiences before going to abstract, be able to do reasoning, make meaning and not forgetting to be able to connect the learning to other disciplines – to name a few.

Figure 1: Real – life experience of dealing with money (concrete learning).

Figure 2: Moving on to symbols.

Reading Reflection – Chapter Two
Exploring What It Means to Know and Do Mathematics
v  The “how”, “why” and “what” of teaching mathematics.
v  It is a science of concepts and processes that have a pattern of regularity and logical order.
My Reflection – During my school days, doing mathematic was in a very regimental form. We were asked to do the sums written on the board and “no questions asked”. At the end of the class session, we submitted our work and teacher would just mark it right or wrong. No proper explanation was given should the answers were wronged and honestly even if I got a few of the sums right – I did not actually know the logic behind it. Having gone through this “meaningless” mathematic experiences, I am determined that the children in my centre should not go through the same process.

What Does It Mean to Do Mathematics?
Mathematics should be for the purpose of their lives skills such as ability to handle money, problem solve, time management, planning and achieving their goals in lives and for effective work performance – just to name a few.  In order to create an environment where children are encouraged to do share and defend mathematical ideas, teachers have to ensure that the classroom is organized for mathematical lessons.

The Language of Doing Mathematics
In the traditional way, mathematic jargons revolved around the words “plussing” and “doing times”. Under the Principles and Standards by National Council of Teachers of Mathematics (2000), the collective verbs describe more of the authentic work of doing mathematics.

These verbs require higher level thinking and involved meaning making and figuring out. It is essential that every idea introduced during mathematics teaching should be understood by the children. By using these verbs, children learning will be more meaningful as they are able to make sense of the concepts taught.

My Reflection – I tried asking my teachers to use some of the verbs (investigate, discover, explore and predict) in replacement to the traditional words in mathematics and observed children’s reaction. It was interesting as initially children were taken aback and took quite awhile to grasp what was actually going on. After awhile, we realized that the children were more relaxed and enjoyed the session more as they felt at ease and were less rigid in their knowledge-acquiring process.

 Picture 3: Children trying to contruct word from the puzzles based on the picture given - thus meeting the objective that mathematics concepts should be applicable to ther disiplines.

Figure 4: From this Percussion activity, children were able to apply some of the mathematical languages such as describe and represent. They were able to describe the different things used as percussions, the colors, shapes and sound it made. And they also learned that simple household things can serve as representation of musical instruments.

Productive Classroom Culture

As stated by O’Conner and Anderson (2003), the teachers would have set a in place a powerful context for student learning should the teachers are able to produce productive classroom culture.

What Does It Mean to Learn Mathematics?
With all honesty, I found it very challenging in my attempt to try out the sums. Exactly as stated in the text, I also questioned myself as to why children should be doing these types of sums – how does these sums help them in their pursuit of knowledge?

Constructivist Theory
The theory revolves around the fact that children are not blank slate, rather, creators of their own learning.  In the assimilation process, new concepts that are introduced will fit in together with the prior knowledge of the child. With this new information, it expands the existing network. As for accommodation, when new concept could not fit in with the existing network, thus the brain has to revamp the existing schemes to incorporate new ideas.

My Reflection – After reading this theory and analyzing figure 2.8, I have a better understanding of how the Constructivist Theory works. Upon reflection, I realized that indeed it is true that we always rely on our past experiences to make connection when we encounter new ideas. And it is also very true that we try to accommodate new ideas and move on when such ideas we have never come across before and we could not connect it to our past experiences.

Sociocultural Theory
A strong believer of Vygotsky, I am especially interested in his Zone of Proximal Development Theory.  I believed through scaffolding, children will be able to maximize their learning potential. Apart from mathematics and other academic components, my teachers usually scaffold children’s learning in other areas as well such as dressing up and packing their own bag independently.
For instance, in the beginning teacher would assist the child in dressing up and packing of her bag after showering. After a while, when the child becomes competent, teacher will slowly move away and child will do it independently.

Figure 5: Once child is competent and able to perform task independently, teacher will cease the scaffolding.

Implications for Teaching Mathematics
Regardless of theories, be it social constructivist or cognitive constructivism, both should be interwoven as active classroom discussion based on children’s own perspectives and problem solving solutions is definitely the foundation to children’s learning. (Wood & Turner-Vorbeck, 2001, p.186, cited in Van de Walle, 2006).

What Does It Mean to Understand Mathematics?
As stated in both sociocultural and constructivist theories, learners make connections of their new ideas to the existing ones. Due to each individual learning ability, learner making connection of the ideas will vary depending on their individual understanding.

My Reflection – In my own word and based on my understanding of the paragraph, the gist of the content is that the way each learner makes connection of ideas will depend on the depth of their understanding. Hence, even if the same concept or idea is posed to a group of children, each of their understanding will be in the form of a spectrum depending on their how extensive or narrow their prior experiences are.

Mathematics Proficiency
v  Conceptual understanding – knowledge about the relationships of a topic or the basic ideas
v  Procedural understanding – knowledge of the rules and the procedures used to carry out mathematical processes. Also, the symbolic representation of mathematics.

Five Strands of Mathematics Proficiency

Benefits of a Relational Understanding
In order to be able to teach effectively, teachers have to put in lots of effort and equipped with the knowledge to educate the children so that they are able to make connections of their learning.
v  Effective learning of new concepts and procedures – the more robust children’s understanding of the concepts, the more connections they are able to built – connecting it to the existing conceptual ideas they have.
v  Less to remember – Knowledge should be in the form of “Big Idea” or as whole rather than isolated concepts.
v  Increased retention and recall – retrieving information is possible when learner has the concept connected to an entire web of ideas.
v  Enhanced problem-solving abilities – in a rich network where concepts are embedded, transferability is greatly enhanced so is problem-solving.
v  Improved attitudes and beliefs – when learner is able to understand the concepts taught and they make sense to him/her, the feeling of “I can do it!” surfaces thus enhancing the learner’s positive mindset towards mathematics.

Conclusion
This reading reflection has taken me to a fairly in depth journey in understanding mathematics. When I was a student, mathematics was one subject that I always had feared on. Even now, though I have somewhat gained better insights to mathematics, I am still apprehensive as to whether I am able to grasp the numerous concepts behind it. With all honesty, I find the textbook to be rather complex and challenging to understand and internalized. Nevertheless, I am full of vigor to learn as much as I can from the upcoming classed so that I will be able to share my knowledge with the teachers in my centre – definitely for the benefit of the children!