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:
- How many dots are on the card?
- How many empty spaces?
- How many dots if we add one more (or less) to make the number shown?
- How many more do we need to mane ten?
- How many less to make five (or any number)?
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.JPGhttp://www.dupagechildrensmuseum.org/images/math_connections.jpg
Suggested 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
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