Through out these six weeks (EAB023), my understandings and beliefs on Early Childhood has developed and changed:
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Chance and Data
Looking at the title for this session, made me wonder; How young kids learn about data and chance? Why do they need to learn data and chance? How can a teacher teach little ones data and chance? The learning experience gained from this session was quite liberating. Three-hours worth of our workshop was really not enough to stop me from thinking about the doubts above. I really felt that I must figure it out myself.
First, I was provided with a box of skittles. The possibility came into play when I need to state ‘which colour has the greatest chance of being selected?’, ‘Is it possible to select 10 orange skittles?’ and so on. Then, in order to answer several questions, I needed to count the skittles in my box. 
While completing the activity, I realized that I used many of the beginning processes mentioned by Irons (1999). The processes include, ‘sorting’ them into coloured group, ‘comparing’ the amount of individual colour and with peers, and ‘ordering’ the individual colour in ascending and descending order. 
Personally, I believe that young children will have a better understanding of the beginning processes when they learn using skittles. Also, I really liked the idea of using a children’ favourite (skittles), in learning the concepts and also indirectly teaching data and chance.
Smith (2001) mentioned that very young children spend time comparing, sorting and classifying everyday objects and they need those concrete experiences with the real world. 
After finishing counting and ordering, I needed to compare my result with my peers. We were required to gather data from other friends and represented the data. We found that the number of skittles in our boxes were not similar; however the average was 38. We also tried to find the mode, median and so on. Curcio and Folkson (1996) suggested that preschool and kindergarten age children experience many exploratory data collecting activities, using their own way of organizing information. It reminded me how an interesting environment can enhance our learning. Also, we were lucky to have some friends videoed our learning experiences using skittles.
In addition, all the data was gathered to include a class chart; using online graphing. NCTM (2000) also claimed that through their data investigations, young children should develop the idea that data, charts, and graphs give information. Personally, I thought graphing was quite hard for young kids. This is supported by Smith (2001) that there are several challenges for young kids; first is one to one correspondence as a key relationship. 
Young kids need to use suitable graphs to enable them in building their one-to-one corresponding. Watson and Mortiz (1999) proposed that it can be in representation such as on pictograph. I also agreed with Watson and Mortiz (1999) that elements of pictographs involving one-to-one correspondence may be counted to tally quantities compared to bar graphs that use scale. 
Thus, pictographs are obviously better than bar graph and other graphs. While graphing in this session, I also realized that by using line graph, young kids cannot make effective connections with the data and they do not see differences.
Which group has more? Which group has fewer?
Or which group is biggest? Smallest? Any there any groups that are the same?
The third level of reading data from Watson and Mortiz (1999), was evidenced when a debate aroused; why was the number of skittles in the box not the same? It is because, when graphing our data on a bar graph on the amount of skittles in individual box, there number varied. We came out with some possibilities and predictions and there was no correct answer as supported by Watson and Mortiz (1999).
Furthermore, NCTM (2000) suggests that young children explore concepts of chance and classroom discussion may focus on events that are certain, are likely to happen, or that are unlikely or impossible. Two games in this workshop gave me a deeper understanding of teaching and learning about chance in EC. First, I played spinner. 
This is aligned with Wall and Posamentier (2007), that teacher should address the beginning of probability through informal activities with spinners or number cubes that reinforce conceptions of other standards, primary number and operations. Children also may talk about the likelihood of an event using a spinner of various colour and may using a spinner of two colours, where there are blue and red (Smith, 2001). I realized that three different spinners provided various numbers of results. It depended on the number of areas or bows. For me, it was quite tricky since I did not know much about the areas. However, it was a challenging activity because we needed to find out ways to make the spinner ‘fair’. Therefore, it may help us while teaching children to note that an imbalance can occur.
My second game was ‘Feeding a Greedy Cat”. 
Here, the dice have different number of faces between red and blue. Thus, the result depended on the faces of dice and there was huge possibility for blue side to win. 
Personally, I thought the use of the word ‘greedy’ was used symbolically to indicate an ‘unequal’ distribution of faces on dice and the possibility that the blue side will win is very high. Interestingly, Smith (2001) indicated that it is natural for children to want to be on the winning side and our society emphasis is that ‘more is better’. 
Thus, the teacher needs to overcome the misconceptions that might arise later on. Lastly, it needs to be highlighted for the students’ self esteem and self-confidence that if you are on the red side (fewer face on dice) you can feel frustrated and disappointed. This is what happened when I throwing the dice.
Referring back on the question which made me wonder; How young kids learn about data and chance? Why do they need to learn data and chance? How can a teacher teach little ones data and chance?,
“Our primary goal as teachers is to equip children with tools they will need to become thinking, responsible citizens on a democratic society, we believe that surrounding them with opportunities to gather, organize and discuss information- and defend it when necessary- involves them in higher levels of thinking, nurtures an investigative spirit and develops competent problem solvers” (Taylor, 1997, p. 149)
References
Curcio, F. R. & Folkson, S. (1996). Exploring data; Kindergarten children do it their way. Teaching Children Mathematics, 2, 382-385
Irons, R. R. (1999). Numeracy in early childhood. Educating Young Children: Learning and Teaching in the Early Childhood Years, 5 (3), 26-32.
National Council of Teachers of Mathematics. (NCTM) (2000). Principles and standards for school mathematics.
Smith, S. S (2001). Early childhood mathematics.
Taylor, J. (1997). Young children deal with data. Teaching Children Mathematics, 4 (3), 146-149.
Watson, J.M. & Mortiz, J.B. (1999). Interpreting and predicting from bar graphs. Australian Journal of Early Childhood, 24 (2), 22-27.
Learning with calculator and games
Previously, in 
I also experienced testing my subitising skills through three gameboards; which are the clown, the egg and teddy race. These games need subitising skills with the dots of dice. I found teddy race was very challenging because I needed to find a good combination to win. I managed to do it even at the first throw, I only got ‘1’ but after that I managed to get ‘6’.
Back in Yet, this week’s lesson has proven (to me) that calculators are not only used to calculate but used as tools to create a fun lesson. 
For instance, I experienced learning skip counting by using calculators. By pressing 3, then plus (+) and then equals (=), we can get 3, 6, 9, 12 and so on. Indeed, calculators can illustrate number patterns for young kids (Smith, 2001). Without our realization, repeated adding of the same number shows a form of early multiplication and repeated subtracting of the same number from a large number
shows a way of thinking about division.
According to
However, I like a little reminder from Smith (2001) that ‘to use calculator accurately, the student must be able to estimate and/or round-off the approximate answer; because the human brains KNOWS how to find a correct answer, NOT the machine” (p.5).
References
Hembree, R., & Dessart, D. (1992). Research on calculators in mathematics education. In J. T. Fey (Ed.), 1992 yearbook: Calculators in mathematics education (p.30).
Smith, S. S (2001). Early childhood mathematics.
http://www.delta-education.com/productdetail.aspx?Collection=N&prodID=3705&menuID=98
Early number sense and basic operations
“All people need to develop a good sense of number, that is, ease and familiarity with and intuition about numbers. This requires a sound grasp of number concepts and notation, familiarity with number patterns and relationships, a working repertoire of number skills and, most importantly, confidence in one’s capacity to deal with numerical situations”.
(Australian Education Council, 1990, p. 107)
I found this quote summarizes my understanding of what a ‘number sense’ is. Before this, I have no idea how a number makes sense; maybe too much time is allocated to solve the mathematics problems instead of understanding the number itself. Not only that, The National Council of Teachers of Mathematics (NCTM) (1989) identified five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers, and meaningful referents for numbers and quantities. Thus, these interrelated components should not be taken for granted as they serve as stepping stone in learning Mathematics in a great level (Bobis, 1996). 
NCTM (2000) added that during the early years, teacher must help students strengthen their sense of numbers, moving from initial development of basic counting techniques to more sophisticated understandings of the size of numbers, number relationships, patterns, operations and place value.
Subitising is also related to number sense as emphasised by Clements (1999) that subitising is a fundamental skill in the development of students' understanding of number. In this session, a ten frame is used to illustrate or capacity to subitise. When required to say the number of ‘dots’ filled the frames,
I realized a few strategies, for example, we can count the empty frame or find a pattern that can produce the number. For instance, my friends and I saw it as a combination of three of threes, two-fours and one, and one-five and one-four. Bobis (1996) also outlined several questions that can be pondered if the ten frame was flashed:
One more thing that I realized was the use of subitising in dice. When we looked at the dice, we tend to take a glance at it instead of counting it one-by-one. Counting is a foundation for students’ early work with numbers (NCTM, 2000; Wall & Posamentier, 2007). 
In this session, some basic understandings mentioned by Unglaub (1997) were applied when we were to count bears. Counting the bears by pointing, touching it or putting it aside was an idea of one-to-one correspondence that children should understand (Unglaub, 1997; Baroody & Wilkins, 1999). My classmates and I were behaving like ‘early childhood students’ in the session, that is; we tended to group the bears according to colour. Not only that, we had a different order of counting, either on the left or right side. Interestingly, it also included basic understandings in counting mentioned by Unglaub (1997) and Baroody and Wilkins (1999) that children normally do.
Watching the video taken in the nursery, gave me some ideas on how to encourage kids count. For instance, the teacher played the music and asked the young ones to move around the class in their preferable ways. Some of them were jumping, standing, and even dancing. It reminded me about the role of physical aspects, kinaesthetic in young children’s development. The use of songs to enhance students’ counting is also an effective strategy for EC teacher (Unglaub, 1997). For instance, the song, “The five little ducks” is using backward number sequence which is a part of rational thinking.
An experience watching videos taken from real classroom (second video) gave me insight about the reality of teaching and learning in EC. 
For instance, the teacher used bears as a concrete objects and related to kids’ life to help them count. The action is aligned with constructivist idea that children recreate or reinvent Maths as they interact with concrete materials, and story problems (Smith, 2001; Montessori, 1976; as cited in Perry & Dockett, 2002). The teacher also sang songs at the beginning and the kids sang after her until all bears fell down. It gave me a clearer idea about teaching EC abstraction and subtraction, and rational counting.
I found this interesting video on counting on and it gave me a clear example of how a child count in his mind.
‘Counting on’, is the most advanced of counting strategies used by children to solve addition problems (Reys, & Lindquist, Lambdin, & Smith, 2006; Young-Loveridge, 2002) However, it is prone to mistakes and ineffective to count a large number. When asking how to solve the questions and help kids develop their thinking strategies, it made me vulnerable to find a solution. At that time, I realized that I cannot explain the process; for example, 9 + 6, but I knew the answer, 15. But, how am I going to teach young kids to solve the question? How am I going to explain to the little kids? This session was really helpful for me in developing my thinking skills and also for my future students. It is because, eventually most children stop using counting on and start using a completely different way of dealing with numbers which involves breaking numbers up and recomposing them in ways that are easier to compute (Young-Loveridge, 2002). Therefore, I found several strategies to answer the questions:
I also noted that number facts can be solved by using more than one strategy. Thus, I think that it is effective to avoid children do exercises that encourage them to use only one strategy.
According to Piagetian theory, adding and subtracting are reversible operations (Waite- Stupiansky, 1999). Reys et al (2006) also proposed the idea that to solve subtraction’s questions, we should think of addition, by encouraging students to think relationship of addition and subtraction. In this session, I watched one video on counting bears backward in a real classroom. Instead of teaching kids, part-whole relations involving subtraction, it involved the use of concrete objects; which was bears and children can move it. I really admired the teacher who at first sang alone, but then the kids followed her and kept singing until all bears fell down. It showed a teacher’s creativity to engage students in counting without their realization. I realized the same backward number sequence also applied in a song entitled, “Five Little Ducks”.
Five little ducks went out one day
Over the hills and far away
Mother duck said "Quack quack quack quack!'
But only four little ducks came back.
Four little ducks…
Three little ducks…
Two little ducks…
One little duck went out one day
Over the hills and far away.
Mother duck said " Quack quack quack quack!"
And none of those little ducks came back.
Old mother duck went out one day
Over the hills and far away
Mother duck said "Quack quack quack quack!"
And all of the five little ducks came back.
References:
Australian Education Council (1990). A national statement on mathematics for Australian schools.
Baroody, A.J. & Wilkins, J.L. M. (1999). The development of informal counting, number and arithmetics skills and concepts. In J. V. Copley (Ed.), Mathematics in the early years (pp. 48-65).
Bobis, J. (1996). Ch 1: Visualisation and the development of number sense with kindergarten children. In M., Joanne. & M. Michael (Eds.), Children's number learning (pp.17-33).
Clements, D. H. (1999). Subitizing: what is it? why teach it?. Teaching Children Mathematics, 5 (7), 400-405.
National Council of Teachers of Mathematics. (NCTM) (1989). Curriculum and evaluation standards for school mathematics.
National Council of Teachers of Mathematics. (NCTM) (2000). Principles and standards for school mathematics.
Perry, B. & Dockett, S. (2002). Ch 5: Young children's access to powerful mathematical ideas. In L. D. English. (Ed.), Handbook of international research in mathematics education (pp.81-111).
Reys, R., Lindquist, M., Lambdin, D. & Smith, N. (2006). Operations: meanings and basic facts. In Reys et al, Helping children learn mathematics (pp.203-232).
Unglaub, K. (1997). What counts in learning to count?. Young Children, 52 (4), 48-50.
Waite-Stupiansky, S. (1999). Games that teach. Instructor, 108 (5), 16-17.
Wall, E. S., & Posamentier, A. S. (2007). What successful math teachers do, grades preK-5: 47 research-based strategies for the standards-based classroom. Oaks, CA : Corwin Press.
Young-Loveridge, J. (2002). Early childhood numeracy: Building an understanding of part-whole relationships. Australian Journal of Early Childhood, 27 (4), 36-42.
Video:
http://youtube.com/watch?v=RpFT330ve74Pictures:
http://www.ed.gov/pubs/EarlyMath/index.html
http://www.nwrel.org/nwreport/2005-03/images/math_inst_kids.jpg
http://www.montessoribc.com/images/100_0397.JPGSuggested website:
Number sense: http://www.learnnc.org/lp/pages/numsense0402-1
Teaching students number: http://funschool.kaboose.com/formula-fusion/games/game_teach_me_1-2-3s.html
Teaching number: http://www.bbc.co.uk/schools/ks2bitesize/maths/number.shtml
Counting: http://www.primaryresources.co.uk/maths/mathsB1.htm
Introduction to Early Childhood (EC) Maths. and Beginning Processes
This first session has given me some quite unpleasant flashbacks of learning Mathematics back in
Reading Irons’ article on numeracy in EC (1999), I found my views, beliefs and understanding of learning and teaching Maths reflected in this literature. 
When it comes to Maths, the first thing that pops into my head is NUMBERS! But, Irons (1999) stressed that numeracy in EC consists of number, measurement, space and shape and data. All these concepts are actually established through and related with children’s learning experiences in play, interactions and explorations of their everyday life.
I really liked the representations of the relationship of concepts and processes as background knowledge of numeracy mentioned by Irons:
Interestingly, this model is also related to some powerful Maths ideas outlined by Perry and Docket (2002). Mathematization, connection, number sense and computation, algebraic reasoning and spatial and geometric thinking, data and probability sense are interweaving with the model in relation to kids’ everyday life. For instance, children use Maths in counting their belongings, such as candies or toys.
Interestingly, kids’ process in Maths learning involved some beginning processes claimed by Irons (1999).
In this week’s workshop, my classmates and 
I were shown some examples of how to apply the process in the classroom by using 3-D blocks. For example; objects are sorted according to number of faces; 
a rectangle has three faces, a pyramid has four faces, and so on. Objects also compared according to the height; a pyramid is higher than a circle. And for patterning, as one of the most complicated processes, we came out with ‘ab aab aaab aaaab’ pattern. Some of my friends suggested the use of hands (clapping), humming and so on. Indeed, it gave me some ideas on how these activities could be used in facilitating children’s learning experiences.
Importantly, teachers need to make sure children are able to identify the ‘likeness’ or different attributes of the objects. It appears as the foundation step before exploring other processes. Children’s recognition of patterns, regularities and common attributes across different objects is also one level of mathematic thinking mentioned by Dienes (1967; as cited in Smith, 2001) which is; ‘generalization’.
Hill (2001) claimed that a conducive environment in learning Maths is very important for young kids. For example; the environment should include hands-on manipulatives that stimulate mathematical thinking, such as sorting, classifying, and problem solving. 
Blocks, games, manipulative toys, and collections of everyday items need to be easily accessible to all students (Hill, 2001).
Thus, I think to engage students more, teachers can use more real 3D objects represent the shape; for example: dice, ball, toy box, ruler, plate, diskette or etc. I also agree with the idea of involving students themselves in translating the pattern in the classroom. Young kids really like to be involved in teaching and learning processes. Once they get the pattern, they will continue the process and solve the problems. My observation in school-based experience illustrated this and the students easily solved it and continued on, no mater how big the number was and how complicated.
When my colleagues and I were required to create a concept map of EC Maths, my group added on some resources that a teacher can used in the strands in QSA Maths syllabus (2004). 
For instance, for time; we can use clocks either digital or analog clocks, for shapes; we can use 3D or 2D objects or visual images, and for chance and data; we can use graphs or charts. However, the learning experience in the workshop gave us some alternatives for the teacher in relation to: song, play, games, rhymes, story, computer and so forth to engage students with Mathematics. For instance, the Internet, students can use online ‘sketch board’ to make their own objects and so forth. What important is teacher needs to make sure the resources are appropriate to their age and level and also have connection to the real world (Perry & Docket, 2002).
References
Hill, B. (2001). The importance of mathematics in early childhood education. Retrieved August 23, 2007, from http://homepages.stmartin.edu/fac_staff/belinda/ece_research/LP.PDF
Irons, R. R. (1999). Numeracy in early childhood. Educating Young Children: Learning and Teaching in the Early Childhood Years, 5 (3), 26-32.
Perry, B. & Dockett, S. (2002). Ch 5: Young children's access to powerful mathematical ideas. In L. D. English. (Ed.), Handbook of international research in mathematics education (pp.81-111).
http://www.delta-education.com/miansplash.aspx?subID=44&menuID=70
http://www.ed.gov/pubs/EarlyMath/index.html
But, what is early childhood refers to?
Perry and Dockett (2002,p.82) outlined several definition;
- The period of a child’s life between birth and 8 years age (C. Ball, 1994; Bredelcamp & Copple, 1997; Organisation Mondiale pour L‘Education Prescolaire, 1980; Schools Council, 1992).
- The first two stages of Piaget’s cognitive development; the sensorimotor stage and preoperational stages. (Piaget, 1926, 1928)
- Children who have been considered lacking in logical representational ability and incapable of using logical and abstract thought, resulting in the perception that children in the early years are “cognitively deficient’’ (Berk, 1997, p. 232)
- A time in which “children rely on increasingly effective mental as opposed to perceptual approaches to solving problems” (Berk, 1997, p. 235)
- Recognised as a vitally important period of human development in its own right, not as a time to grow before ’real learning’ begins in school” (Bredekamp and Copple, 1997, p. 97)
And how do children learn Mathematics??
Reys, Lindquist, Lambdin, & Smith (2007) mentioned that the vision for mathematics education promoted by the National Council of Teachers of Mathematics (NCTM, 2000) is for all children to learn mathematics with understanding.
- Learning comes from experience and active involvement by the learner (John Dewey)
- Learners actively construct their own knowledge (Jean Piaget)
Reys, R., Lindquist, M., Lambdin, D. & Smith, N. (2007). C 1: School mathematics in a changing world; C 2 : Helping children learn mathematics with understanding. In Rey et al, Helping children learn mathematics (pp. 1-36).
Perry, B. & Dockett, S. (2002). Ch 5: Young children's access to powerful mathematical ideas. In L. D. English. (Ed.), Handbook of international research in mathematics education (pp.81-111).
Wall, E. S., & Posamentier, A. S. (2007). What successful math teachers do, grades preK-5: 47 research-based strategies for the standards-based classroom. Oaks, CA : Corwin Press.
Pictures
