Indeed, through out this unit, I have learnt so many things that really constructed my beliefs and understandings on Mathematics in general and in Early Childhood context in specific. Here are some of the lessons that really given me new insights and developed my personal philosophy:
Therefore, I believe that the teaching strategies used in this unit have a resemblance with Bredekamp and Rosegrant’s (1992) learning cycle to early childhood education. Awareness, exploration, inquiry and utilization are four repeating processes encompassed the learning cycle for young children.
As an example, I came to this unit with some recognition of concepts that developed from my experience learning Maths, then, I constructed my own understandings and meanings of the concepts while participating in the workshops using ‘sensory experiences’ with materials, classmates and new concepts. Learning in this unit in another culture (Australian culture) was really giving me a ‘medium’ to compare with Malaysian thinking skills and Maths teaching, as well as we recognized other cultural thinking. Therefore, towards the end of the unit, I designed some resources which were Australian based to be used in Malaysian classroom later on. The processes really demanded my understanding of the concepts to be used in another contexts, settings and situations. In all, this unit gave me new perspectives of Mathematics teaching and learning processes.
Another significant insight that contributes to my personal philosophy is related to awareness of the ‘gap’ between home and formal mathematics classroom. Before came to this unit, my perception of teaching was shaped by traditional ideology that ‘students come to class with empty vessel’. A research done by Aubrey (1997) also proved the same, ‘teachers seemed quite unaware of the rich, informal knowledge brought into school’ (p. 85). I have learnt and observed through out this unit that children are surrounded by Maths. Some of resources used were quite familiar resource that apart of children’s toys, such as clay, cubes and so on. Indeed, everyday, children are bombarded with new surprising and exciting experiences whilst attempting to make sense of previous experiences, therefore, they can become quite competent mathematicians long before they enter formal schooling (Bottle & Alfrey, 1999). In short, I need to make sense of the students’ basic mathematical knowledge on previous experiences align with the long term mathematical learning outcomes.
Being a reflective practitioner is always important for making the teaching and learning process a success. Having reflected on the learning experiences gained from this unit, I found the key to critical reflection is the notion that knowledge is created and constructed, rather than a received commodity (Ailwood, et al, 2006).
In conclusion, my personal philosophy is more to give children value in mathematical thinking that enable them to think independently with enjoyment and engagement therefore to develop positive attitudes towards mathematics.
References
Math Resources: Make and Take
This week’s workshop has really made me feel closer to the teaching profession, rather than being a ‘early child’ as before. I was introduced to many Maths resources that were very useful in teaching Maths to young kids. Overall, in this workshop, my classmates and I shared ideas for Maths resources. Then, we made them using all the materials provided and took the resources with us. So, I was required to do Maths resources that were applicable and suitable for early childhood age.
Resources to teach number sense
Firsty, I was provided with some resources that can be used to teach number sense; which were hundred charts, empty number lines, and ten frames. These three types of resources will become a useful tool for me in introducing more hands-on activity while teaching Maths in
The box of facts
Then, I was introduced with ‘the box of fact’ to teach addition, subtraction, division and multiplication. Indeed, it was my first time experience learning using that amazing toolkit! I was amazed by these sorts of things that young kids here have opportunity to have in learning Maths. It was totally different compared to my previous schooling in 
“The box of facts to multiplications and division”. Indeed, it would help young kids to learn multiplication and division in more enjoyable ways, not only memorizing the number facts! When my lecturer demonstrated how to use it, I was really amazed how systematic it has been organized. In order to answer the question, for instance, 6x8, students can count the black dots and it also similar to 8x6.
In this workshop, my friends and I tried to construct one card similar to the one in the box of fact. We were using a hard A4 paper and using some stars as ‘the black dots’. First, we needed to organize where we should fold the paper and pasted the stars. Then, we decided to use 4x4 as the basis of the card. That’s mean, if we wanted to do 2x4, we needed to fold it until it had two stars per row. Then, we wrote down the operations on the top of it horizontally. It was because, we were told that, in 
I thought it was an effective way as young child would read the operations align with the dots. It was shown in the sample that we used as a guideline. But, I realized that one of my friends wrote it quite differently. She started with the number of columns first, not the number of a row. I wondered how it will affect young kids in solving the operations. After discussing, we found that we still can use it as an extension activity for kids. It is because, to solve the operations, they can’t simply rely on the number of stars in a row.
Ladybird sixes
Another engaging yet educational resource was a ‘ladybird sixes’. 
This interesting stuff would apply part-whole relationship for ten. That’s mean it should been provided with two numbers (part) that equals to ten (whole). I had listed “1 and 9”, “2 and 8”, “3 and 7”, “4 and 6” and “5 and 5”. While doing this task, I realized that there were lots of different ways to put six spots in each ladybird.
Interestingly, according to Resnick (1983; as cited in Young- Loveridge, 2002), this interpretation of numbers in terms of part-whole relationships enables children to think about numbers as made up of other number and help them in achieve major conceptual in early years.
My friends and I did it using coloured paper, colourful self-adhesive, markers, scissors and glue. At first, we cut coloured papers into oval shapes. We decided to cut five different colours as we wanted to do five ladybirds. We also divided its wing cases into two. Then, I put the colourful spots using colourful self-adhesive on the two halves of the ladybird’s back. Then, my friend put its eyes and eyelashes to make it more ‘real’.
But, how could we use it? At that time, I tried to be a student and I wrote numbers that could makes six. 
By subitising, I ‘added’ the spots on the left wings with the right one. What I have done was I used addition sum by counting the spots on the ladybirds’ back. However, there was a reminder from my lecturer that we can’t use the word, “makes” to explain the operations. We were quite confused especially with the native speakers who frequently used it, instead of ‘equals’ or ‘same as’. Then, we were told that the word “makes” will become meaningless when it comes to algebraic thinking. For instance; 6+2 makes 8, but 8 can’t make 6 + 2. Therefore, I realized that as a future teacher, I need to watch my language use in explaining operations or processes in Maths.
Monster
In this workshop, my friends and I also have an opportunity to create our own monster to teach students about counting backwards. We used paddle pop sticks, pipe cleaners, strings and other materials such as coloured paper, stars, colourful 
self-adhesive labels to make monsters. We decided to make ten monsters so that we could use the book, Monster Math by Grace Maccarone and sing:
Ten hairy monsters
Went to the park,
One skipped away,
Now there ware nine…
The process of ‘monsters’ making was really demanding bth patience and creativity. It is not easy ‘putting’ the pipe cleaners around the paddle pop sticks. We needed to ensure that monsters were similar and have some distinguish features so that young kids can differentiate it. Indeed, wonderful ideas by using these ‘monsters’ can engage young kids in learning Maths; especially on counting.
Shape puzzles
“It is quite hard to do”
“I can’t use knife properly as I may cut my fingers”
“Why it is become complicated as we need to cut it, paste it and cut it again?”
Here were some comments and mutterings while my friends and I were given some options to do shape puzzles. 
Finally, only one of us decided to do it. Here were some processes that I observed:
1. Select three large pieces of thick cardboard
2. Using a craft knife, she cut a triangle out of the first piece, a square out of the second piece and a rombus out of the third piece.
3. She glued the left over shape (after cutting the shapes) on coloured papers.
4. She cut the coloured paper which had been glued.
5. She made a handle for each cut-out shape by winding some pipelines.
I
ndeed, all we need were patience and ‘visualization’ of the shapes that will become. Then, we had to take turns fitting the cutouts into the matching shaped holes. It was enjoyable and I thought young kids would love it, as they will satisfy when they are able to insert the shape into its holes. It is because; “teacher carries the responsibilities for adjusting the level of difficulty so that each child experiences not only a challenge but also a sense of success” (Schwartz, 1995, p.416). The process involved was not only a single orientation but also one to one correspondence as one hole only suit with one shape.
references
Designing learning experience
For this learning experiences’ workshop, my friends and I chose to work on sh
ape topic by designing a learning experiences focusing on the 3D shapes. We used the book, ‘Wyne’s New Shape’ by Calvis Irons and Peter Shaw. Basically, the story is about Wyne, a lumpy piece of wood and he did not have a smooth and neat shape like his friends. He was very sad until he found a wood shop. 
There, Wyne was transferred into a new shape. He has been rectangular, then cylindrical and finally a round ball. To make learning more meaningful, we decided to use some 3D cubes available.
Firstly, we planned the lesson. We constructed some questions to be asked while reading the big book. The minimum of question was one per page. Here were the questions:
1. Look at the cover, can you tell how many shapes is there? Which one do you think is Wyne? Why is this guy frowning? Do you think this story is a sad story or a happy story?
2. Can you identify which one is Wyne now? Can you count how many shapes are there?
3. What do you think that shop is that? Why do you think this shop is amazing to Wyne? Can you guess what will happen next?
4. What do you think the owner of the shop is doing?
5. What shape is that? Why does Wyne feel great? Can you find one shape here that is similar to Wyne now? (Students need to pick up 3D shapes provided). Anybody knows the name of this shape?
6. What shape do you think Wyne now? What will happen to Wyne now?
7. Can you find a shape that is similar to Wyne now? Anybody knows the name? Can you find an object that is similar to Wyne in this classroom?
8. What shape do you think Wyne now? Which shape is similar to Wyne now? Why do you think he is so happy?
We also thought of several extension activities;
- find similarities and differences between these shapes
- find similar objects at home
- do shape walk
- play dough- transformation of Wyne’s shapes.
Then, we conducted the activity with a group of students. We put them on the carpet in a circle and they could see each other. They needed to act as four to five year old children. One of us read the story and another one was asking the questions. And another member was taking photographs of the lesson conducted. At that moment, I felt that everything went smoothly, but in the middle some of them lost attention. They started yawning. They tended to be passive and not respond to the questions asked.
At that time, I was the photographer and I observed my friends’ decisions regarding this matter. She decreased the number of questions per page and it still gave time for interacting with the ‘students’. From my observation, I found that students like the open ended questions such as asking for their opinion and also ask them to find similar objects, in the classroom.
We were lucky to have some feedback on our ‘lesson’. First, the story itself, it was 
a good story to teach students about 3D shapes. It also had a good moral value for students to have a strong will and never give up in their life, like Wyne did. Students also experienced touching and feeling the shapes itself while Wyne had transferred into new shapes. This activity gave them ‘real’ experience of 3D shapes (Wall & Posamentier, 2007). Students also had a chance to relate it to their real life and real environment as they were required to find similar objects in the classroom.
However, some of the ‘students’ claimed that too many questions were asked and it hindered their attention and focus on the story. One friend of mine said that she can only focus in the middle and towards the end, she got lost and daydreamed. She didn’t have any idea why and what happened. She related it to the teacher who read the book, she said she tended to pause too long and kept repeating the questions.
Therefore, in the future, I think that I as a teacher need to be aware of young children’s attention span. Once they got bored or lost, they tended to distract other students and misbehave. One of my classmates suggested that we read the story first and the second time, we ask students questions. But, for me, it may hinder a suspense that we are going to create from the story. So, it depends on the story itself.
Another good lesson that I have learnt from another group was giving clear instructions. One of their ‘teachers’ confused us while dividing us into groups. It caused chaos and we ended up wasting a lot of time and therefore interest before we sat in our own group. Indeed, the teacher needs to give clear instructions especially for young children who need this.
In short, I found that this week workshop not only about designing learning experience to be used, but it actually a learning experience. It gave me some insight related to real teaching and the learning process in a classroom.
reference
Pattern, Algebra, Pattern Pattern, Algebra
This week’s workshop brought back my secondary school memories of algebra, functions and pattern. I have learnt algebra and functions in my secondary school through the use of formula in finding the ‘unknown unit’. 
Frequently, what I have learnt was memorizing the formula from the textbooks and used it to solve the mathematical problems! Therefore, this ‘mindset’ has governed my initial understanding of how young kids learn pattern, functions and algebra. One big question in my mind was; do we expect young kids to solve algebraic equations and function problems?
Algebra
The journey towards my confusion as well as curiosity began when we are needed to share some thoughts and feelings that pops into our heads when we heard the word, “algebra”. 
The list of related words was a relief as some of them are very familiar to me. Taylor-Cox’s (2003) article also gave me clearer idea on what algebra in Early years looks like. I noticed the word ‘pattern’ was on the top on the list. But, how are patterns connected to algebra?? How can repeating, growing and relationship patterns help young kids learn algebra?
Pattern
However, one more thing that I have learnt from this workshop was patterns are not just merely the repetition of ‘red bear, blue bear, red bear, blue bear’, but it more than that. 
Patterns are a way for young children to recognize order and to organize their world and are important in all aspects of math at this level (Clements, 2004). As claimed by Economopoulus (1998); Heirdsfield (2007); Smith (2001) and Taylor-Cox (2003), pattern involves recognizing, describing, extending and translating patterns. The video shown in this workshop also explained the real-situation where young kids create patterns by sorting objects, putting number to the pattern, experimenting using the number and saying the number.
From the video, I can relate to ‘number sense’ that have been learnt in Week two; where through patterns, students can do split counting of multiplication. Indeed, through patterns, we can predict pattern, what comes next? What comes after AAAABBB?? Here, then through finding ‘the missing pattern’ I was be able to relate to algebraic thinking! Indeed, through repeating and growing pattern, I can relate to find the missing pattern and therefore find the solution!
Functions
Instead of pattern, functions also play an important role in algebra in early years (Taylor-Cox, 2003) and
Purple cat -----à black cat
Pink dog ------à green dog
. Therefore function= Purple ---à black
Pink ----à green
Indeed, the situation became harder when one of my classmates suggested ‘red bird’. It made me wonder what colour of ‘bird’ it will become. Personally, I found this activity can result in confusion, but indeed it was a good example; as we can’t predict pattern unless we used some rules? How can it serve as a function (qualitative change)?
In addition I found the function machine was an interesting way of introducing children to generalization of patterns and functions. The activity of using a function machine in ‘connecting’ the relationship between two objects was really gave me an idea how it is ‘functioning’. However,
Generalization
Algebraic thinking is also related to generalization. 
A ‘swimming pool’ activity in this workshop has constructed my understanding of ‘looking for generalization’. Indeed, I needed to use visual thinking as it used white tiles and blue tiles. After I built the pools using the guideline provided, I needed to identify the patterns from the blue tiles and the white tiles. I documented my ‘thinking’ as I needed to predict; if there are 49 blue tiles, how many white tiles are there?? Is there are 44 white tiles, how many blue tiles are there? Here, I realized that I need to construct a formula so that I would not count them on to solve the problems. I needed to look for generalization!
Equality and inequality
Another important feature of algebra in early years is equality and inequality (Taylor-Cox, 2003). Equality or sameness is an important concept as it offers children with recognizing, defining, creating and maintaining equality. Through this workshop, I realized that ‘the concept of balance’ is important in solving algebra equations! Using a pan balance scale, the activity has lead me to 
understand equality as balance. As claimed by Charlesworth and Kind (2007), by exploring with a pan balance, children find that if they put certain objects on each side, the pans will balance. Not only that, to trigger my thinking, I was required to think of the way mathematical problems were solved using arithmetic and algebraic.
Personally, I found that using arithmetic thinking was easier as I used to do it in number facts. I tended to rely on my ‘counting ability’ when I needed to use algebraic solutions. Even though I tried not to count it, it seemed automatic. However, on the positive side, I found that this strategy of thinking is effective for teaching young kids at the beginning, before teaching them how to count big numbers and memorise the number facts!
Final thought
Therefore, through the activities in this workshop, I agreed with NCTM (2000) that algebraic concepts can evolve and continue to develop; although the concepts discussed are algebraic. Furthermore this does not mean that young children are going to deal with the symbolism often taught in a traditional schooling, because before formal schooling they have already developed beginning concepts relating to patterns, functions and algebra in their everyday lives.
My personal philosophy
Exploring this topic has brought me some insights. From this workshop, my personal philosophy is constructed around, ‘it is never too young for kids to learn about pattern and algebra. Indeed, it is not as I thought earlier that; the young kids need to solve algebraic equations and function problems while learning algebra and pattern, but children can learn algebra through patterns. I agree with Charlesworth and Kind (2007) that algebra begins with a search for patterns. By identifying patterns, it helps children be able to make generalizations beyond the information available and therefore serves as an important component of the young child’s intellectual development in algebraic thinking. They need to be invited to look for the patterns that are all around them in their everyday world because pattern spotting and looking for rules is fundamental and at the heart of mathematics. Therefore, interesting learning experiences will help young children learn something more than traditional people did that is; learning algebra in year Six!
References
Measurement
“A four year old boy was rolling out a piece of clay into the familiar snake form. As he finished, he repeatedly moved his hands from the centre toward the ends on the rolled-out clay object. He was quietly saying in a measured tone, “Long. Lo-o-ong. L-o-o-o-ng”.
Schwartz (1995)
This is just one of the scenes that is played out again and again in pre-kindergartens and kindergartens across the world. Indeed, I also experienced the same when I was in kindergarten. As far as I can remember, once I was required to make a long snake, I tried my best to make it longer than before. In addition I was using my thumb and my index finger to measure it. However, when I tried to use a ruler to measure it, I was told that a ruler can measure length, but I can not measure ‘my snake’ with it! It was hard and I got confused. Eventually, I used my preferred unit; my thumb and my index finger. After reading Schwartz (1995) article, I realized that I am not the only one. It is amazingly common! Young children in kindergartens use repeated units and both standard and nonstandard unit as I used. They also spontaneously solve measurement problems emerging from their ongoing activities and eagerly discover measurement relationships as they construct the materials (Schwartz, 1995). This situation also related with five stages of development advanced by Piaget (Charlesworth & Kind, 2007): at the preoperational stage; children develop the measurement stage of making comparisons and in transitional to concrete operational, they tend to use arbitrary units.
Interestingly, Board of Studies NSW (2002) mentioned that students develop the key understandings of the measurement process using repeated informal units through these five principles:
1. The need for repeated units that do not change
2. The appropriateness of a selected unit
3. The need for the same unit to be used to compare two or more objects
4. The relationship between the size of the unit and the number required to measure, and
5. The structure of the repeated units.
Teaching that builds on students’ intuitive understanding and informal experiences helps them understand the attributes to be measured as well as what it means to measure (NCTM, 2000)
I agreed with NCTM (2000); learning about measuring in this week’s workshop put me in a comfort zone as I have some prior and background knowledge of some aspects of measurement: capacity, volume, area, length, height, mass and temperature. Not only that, the beginning process advocated by Irons (1999): identifying, simulating, comparing, sorting and ordering are still used as a teaching sequence. In the three hours workshop, I learnt many new aspects of measurement.
Length
In everyday life, children measure the length of everyday objects such as books, boxes, toys and pencils with non-standard units (Smith, 2001). Indeed, they measure things for their own purposes, which most often include using the information to make comparisons, to describe their constructions, and to make objects fit or align. If the child is not interested in why he or she is being asked to find out the length of an object, the task lacks purpose and fails to engage the child meaningfully (Schwartz, 1995). In this workshop, we were required to measure height using the non-standard units. Interestingly, my friends and I used the methods for measuring for physical quantities that agreed with most professionals (Smith, 2001). They are:
1. Choose the appropriate unit
2. Use the unit to cover object with no spaces or gaps
3. Count the units
4. Decide on what to do with leftover parts
For the non-standard units, we decided to use unifix cubes instead of paper clips, streamer and paddle pop sticks. We preferred to use colourful unifix cubes as we can combine it while measuring and it was fascinating and colourful.
Before measuring, we estimated and chose the highest person to be our ‘guideline’. While measuring, we made sure that it was end on end with no overlap. Then, we counted the unifix cubes, and we used the highest person’s height as a ‘standard’ unit. Then, while measuring others, we put aside the leftover parts and documented our heights.
After finishing recording our height, I found that this activity was a new and challenging activity for me. As an adult, I myself felt quite ‘unsatisfied’ with the result and the process of measuring. It was challenging and involved lots of processes.
So how will young kids cope? Curry, Mitchelmore and Outhred’s (2006) article really gave me new insight. They stated that there is strong evidence that, in the early stages of learning about measurement, children do not see the need for congruent units when measuring individual objects (Bragg & Outhred, 2001; Outhred & Mitchelmore, 2000). Several studies have revealed students’ difficulties with the principle that the object being measured must be covered with no gaps or overlaps (Bragg & Outhred, 2001; Curry, Mitchelmore, & Outhred, 2006; Outhred & Mitchelmore, 2000). Another issue arose from the activity; the accuracy of the units used.
If we were to compare the accuracy of using paddle pop sticks and unifix cubes, I found that using unifix cubes was more accurate. It is because, unifix cubes have smaller components and the probability of an error occurring was small as we used different cubes. But, with paddle pop sticks, normally we used the same stick and it has quite a long ‘standard’ per stick.
Time
Scott (1998): time= space, time as a domain is also measurable
I agreed with Scott (1998) that time equals to space. According to Heirdsfield (2007) and Smith (2001), the concept of time involves time sequence of events, duration of events and duration of various units of time. Everyday classroom activities provide opportunities to estimate time. In this workshop, I experienced measuring time by estimating how long a pendulum swings to construct a unifix tower of 10 cubes. 
After measuring the time for 10 unifix cubes, we added the number of unifix cubes with 12 and 15. From the table, I found that there was no change in the time in constructing the 10 and 12 unifix cubes. One of us suggested that it just involved the addition of 2 cubes, so it wasn’t so much. But, it changed with 2 or 3 swings when they were required to construct a unif
ix cube of 15 cubes! It made me think of the reason. I believed that something was missing or maybe some mistakes had occurred. My hypotheses was that my friend had miscalculated the unifix cubes or she may have been having difficulty with it at that time. I agreed with Kribes-Zaleta and Bradshaw (2003) that children actually have clear ideas about what it means to use units in measuring length, but the incompleteness of their understanding of the formal properties of measurement units makes this exploration interesting.
Mass
Another interesting and educational activity in this workshop was using paddle pop sticks to estimate the mass of a variety of objects. We used a balance scale and paddle pop sticks to measure the mass. Personally, I felt very excited to know how paddle pop sticks can be used as a non-standard unit and estimate the mass of other objects. Here is 
the result of mass of a variety of objects . 
I think this activity was a good idea for teaching young kids as mentioned by Smith (2001) that measurement activities must involve ideas that they can enjoy and that have significance for their lives. Not only that, a foundation in measurement concepts enables students to use measurement systems, tools, and techniques should be established through direct experiences with comparing objects, counting units, and making connections between spatial concepts and numbers (Wall & Posamentier, 2007).
Final thought and my personal philosophy
In short, as a future teacher, I found a useful reminder from Wall and Posamentier (2007):
[the] teacher can guide young children’s exploration and understanding of measurement concepts and relationships by making resources for measuring available, planning opportunity to measure, and encouraging children to explain the results of their actions.
This strong reminder also constructs my personal philosophy in teaching Mathematics. I should not merely depend on drilling and memorization as to ensure my students learn Maths and pass the exams. I should use myriad of resources, as I used in this workshop to help them build their understanding and knowledge of Mathematics. I also should give my students chance to measure things in their environment and therefore relate it to meet the needs of curriculum. I also should not ignore their ideas and thinking of measurement. “The best time to learn mathematics is when it is first taught; the best way to teach mathematics is to teach it well the first time” (National Research Council 1989, as seen in Cockcroft and Marshall, 1999:p. 329).
References
References for picture
http://www.chadiscrafts.com/fun/Claysnakes.jpg
Suggested website
>> Heaps of resources that can be used to teach measurement for young kids. Interestingly, it has been divided into aspects of measurement such as length, time and so on. It's also easy to access in any formats such as powerpoint presentation, flash player, or pdf files.
This blog is exclusively designed to complete my EAB023 (Maths. Explorations in Early Childhood) assignment. Thus, it is solely for academic purpose and also for my reflection on teaching and learning processes. Therefore, it should not be taken for absolute reference in any ways.
Any queries or comments:
n3.ismail@student.qut.edu.au
