References and Bilbiography

References and Bilbiography

Bobis, J.; Mulligan, J.; and Lowrie, T. (2008). Chapter 9: Promoting Number Sense: Beyond Computation in Bobis, J.; Mulligan, J.; and Lowrie, T, Mathematics for children: challenging children to think mathematically, Frenchs Forest, NSW: Pearson Education Australia, pp.215-242.

Dockett, S. & Perry, B. (2002). Chapter 5 : Young Children's Access to Powerful Mathematical Ideas in English, Lyn D (ed), Handbook of international research in mathematics education, Mahwah, NJ: Lawrence Erlbaum Associates, pp.81-111.

Irons,R.(1999). Numeracy in early childhood. Educating young children: Learning and teaching in early childhood, 5(3), 26-32.

Perry, Bob and Dockett, Sue. (2002). Ch 5 : Young Children's Access to Powerful Mathematical Ideas in English, Lyn D (ed), Handbook of international research in mathematics education, Mahwah, NJ: Lawrence Erlbaum Associates, pp.81-111.

Reys, B. J., & Barger, R. H. (1994). Mental computation: Issues from the United States perspective. In R. E. Reys & N. Nohda (Eds.), Computational alternatives for the twenty-first century: Cross-cultural perspectives from Japan and the United States (pp. 31-47). Reston, VA: National Council of Teachers of Mathematics.

Reys, R et al. (2009). Chapter 9: Operations: Meanings and Basic Facts in Reys, R et al, Helping children learn mathematics, Hoboken, NJ: John Wiley and Sons, pp.184-212.

Steiner (2009).Montessori Answers. Retrieved September 2, 2011, from http://www.montessorianswers.com/progressives.html

Yelland, N. (1999) Technology as Play. Early Childhood Education Journal, 26(4), 217-220. Retrieved from: http://www.springerlink.com.ezp01.library.qut.edu.au/content/prp0320124010330/fulltext.pdf

Picture References

[Untitled photograph of 100 chart. Retrieved September 5, 2011, from http://www.abescastle.com/Images/100%20Chart.jpg

[Untitled photograph of dig game. Retrieved September 5, 2011, from http://www.fun4thebrain.com/Division/dinoDigDiv.html

[Untitled photograph of tweet tweet game. Retrieved September 5, 2011, from http://www.ezschool.com/Games/Addition.html

[Untitled photograph of simple subtraction. Retrieved September 5, 2011, from http://www.primarygames.com/takeaway/question_2.htm

[Untitled photograph of counting game. Retrieved September 5, 2011, from http://www.pitara.com/activities/math/counting/index.asp

[Untitled photograph of washing line game. Retrieved September 5, 2011, from http://www.curriculumsupport.education.nsw.gov.au/countmein/children_washing_line.html

[Untitled photograph of penguin game.Retrieved September 5, 2011, from http://multiplication.com/flashgames/ConeCrazyLevels.htm

[Untitled photograph of houses. Retrieved September 5, 2011, from http://www.learner.org/courses/teachingmath/gradesk_2/session_04/images/s04_img_house.gif

Reflection

Workshop One :

I view mathematics as one of the subject in school where it involves numbers and calculation. But, it is actually more than just a subject. It is something that revolves around us daily, without us knowing that. Almost everything requires mathematical solution to work. Young children need to immerse themselves in the process by doing their own activities with little help from the teachers. Yelland (1999) supports the principle of Froebel and Montessori that both of them seemed to agree that children learn most effectively from self-directed activities which give them a high level of empowerment and ownership and thus, high intrinsic motivation.

Learning mathematics is not only restricted to pen and paper. It can be extend to another way of learning which is by play or using realia to improve the understanding of mathematics with young children. According to Yelland (1999), early childhood educators have always contended that young children learn best through play. It is not all about numbers and symbols as household items could be used to further enhance the learning. Balls could be use as an object to count instead of the normal numbers on chalkboard.

My assumptions about early childhood mathematics are that the young children will only need to know the basic of mathematics which is to know about the numbers and the simplest of addition, subtraction, multiplication and division. This will help them to slowly develop the skill to enable them to be involved in the learning and eventually help to increase their knowledge regarding mathematics.

Workshop Two :

I learn to count in mathematics during my primary years of schooling thanks to my lovely mathematics teacher. Solving problems in mathematics does not restrict to one way only but it can be done in several ways through computation. In computation you have either mental or written computation. I am more familiar with written computation because I was taught to do the written rather than mental computation which involves the tools that you normally have with you, that is pen and paper.

I was taught to use the written formal algorithm where I would have to solve the numbers aligned on the right first then moves on to the one on the left. If it was involving bigger numbers, I would first draw stick figures to indicate that is the number that I want to represents. It is easier this way because you could count the stick one by one instead of seeing the big numbers who young children find a bit difficult to digest. As the year goes by, a new method was introduced which is the number line. This is a good technique for young children and I myself loves this technique as it its fast and you could do addition subtraction, multiplication or even division with it!

Learning multiplication and division is quite similar to learning addition and subtraction. Teacher used models as to ease the students' understanding about the concept of multiplication and division. I think that it is good because the students could see the objects and were able to solve it rather than they have to imagine the numbers involved in their head. Also, I have to memorize the multiplication tables from 2 - 12 every single day during maths hour. This is to ensure that we are able to subitize the numbers to be multiplied to further deduce the steps in order to crack the answers.

In conclusion, during my primary years I was encouraged to use both written computation and mental computation by my teacher. To solve mathematical problems it does not depends on how you try to solve the problems but as long as you get the correct answer, you are on the right track. One of my maths teacher used to say that to encourage us in class.

Workshop 2

Number and Operation

Subitizing

Subitizing is “instantly seeing how many” or in other words is the direct perceptual apprehension of the numerosity of a group (Clements, 1999). Unlike counting which focuses on the unit, subitizing focuses on the whole and unit and it underlay the number idea. There are two types of subitizing :

Perceptual Subitizing

Recognition of number without using other mathematical processes (Clements, 1999)

Conceptual Subitizing

Recognition of the number pattern as a combined part and as a whole (Clements, 1999)

Young children will eventually able to guess the number of block being placed on the paper correctly without having to count each block using subitizing.

Example of subitizing game, the children would have to swat the fly using the fly swatter as the number on the fly appears twice. It is similar to the ‘snap’ game. This will enhance the subatizing skills of the children.

Clement (1999) says that students can use tens frames to visualize addition combinations as it will help the student to recognize the number and use the model in calculating sums.

‘Teddy bear race’ activity is one of the activities that use the tens frames to visualize the simple addition. The roll of the dice indicates which teddy will move one step closer to the goal. It needs 7 moves altogether for the bear to reach home. It is a simple addition of 1+1+1+1+1+1+1=7.

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Counting

According to Gelman & Gellistel (1978), there are five principles govern and define counting involved in early childhood mathematics:

One-to-one principle

-This involves the assigning of one, and only one, distinct counting word to each of the items to be counted.

Stable-order principle

-To be able to count also means knowing that the list of words used must be in a repeatable order.

Cardinal principle

-The final object in a collection represents the number of items in that collection.

Abstraction principle

-Counting tangible or non-tangible objects. children need to appreciate that they can count non-physical things such as sounds, imaginary objects or even the counting words.

Order-irrelevance principle

-This principle refers to the knowledge that the order in which items are counted is irrelevant as long as the objects in the collection are counted once and only once.

Easy game on counting objects - http://www.pitara.com/activities/math/counting/index.asp

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Number word sequence

Forward number word sequence

Forward number sequence starting from 1 – 10

Number word after

Q : What comes after 1

A : 2

Backward number word sequence

Backward number sequence starting from 24-15

Number word before

Q : What number comes before 16 ?

A : 17

http://www.curriculumsupport.education.nsw.gov.au/countmein/children_washing_line.html

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Number Rhymes

Simple song that helps young learner to count numbers in correct order by singing.

Number Rhymes I

One, Two
I go to school
Three, Four
I play outdoor
Five, Six
I read my books
Seven, Eight
I am never late
Nine, Ten
I sleep by then.

http://www.kidsfront.com/rhymes/number_rhymes.html

Number Rhymes II

One for the money,
And two for the show,
Three to make ready,
And four to go.

http://www.kidsfront.com/rhymes/number_rhymes.html

Reys (2009) says that addition and subtraction are inverse operations; that is, one undoes the other:

5 + 8 = 13 ---> 13 – 5 = 8 ; 10 + 15 = 25 --> 25 – 10 = 15

Enable children to learn addition effectively using simple addition http://www.ezschool.com/Games/Addition.html

Simple subtraction game

http://www.primarygames.com/takeaway/question_2.htm

Multiplication can be viewed as repeated addition:

6 X 3 = 18 --> 6 + 6 + 6 + 6

An online multiplication game, children will find this to be an interesting game of multiplication as it involves cute character and ice cream.

http://multiplication.com/flashgames/ConeCrazyLevels.htm

Division can be viewed as repeated subtraction:

32 ÷ 8 = 4 ---> 32 -8 -8 -8 -8 = 0

Dig It! is an example of division game that requires children to do a quick division to get the bone in the ground.

http://www.fun4thebrain.com/Division/dinoDigDiv.htm

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Number Facts Strategies

There are several number facts strategies that children use for addition and subtraction which are:-

Count on from larger:

9 + 50 = 50, 52, 53, 56,…59

11 + 27 = 27, 28, 29, 30,…38

Doubles:

3 + 3= 6

10 + 10 = 20

Near doubles:

12 + 13 = (12 + 12 + 1) = 25

24 + 25 = (24 + 24 + 1) = 49

Halving:

24- 12 = 12

18 – 9 = 9

Near halving:

27 - 13 = 14 = (26 - 13 = 13 + 1= 14)

17 - 8 = 9 = (16 - 8 = 8 + 1 = 9)

Bridging to 10, 100 or nearest decade or multiple of 10:

59 + 21 = 60 + 20 = 80

89 + 91 = 90 + 90 = 180

Addition and Subtraction

Tens frame could be used to teach young children about addition.

There were 3 people sitting on the ground, 3 came in, how many people were there now?

By using objects with different colour to show each part of the addition, young children will quickly grasp the simple addition by adding group of the colour separately, 3 + 3 = 6 .

Same objects can be used for different operation such as subtraction, there were 6 people sitting on the ground, 3 went away, how many people were there now? 6 – 3 = 3

Two tens frame can be used if the numbers involved in the addition is greater than 10

Example from the picture, 7 white blocks + 5 blue blocks = 12 blocks

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Mental computation strategies

Docket and Perry (2002) claim that mental computation is one of the parts in young children's learning about number. It can be used as a tool to facilitate the meaningful development of mathematical concepts and skills and to promote thinking, conjecturing, and generalizing based on conceptual understanding (Reys & Barger, 1994). It includes the ability and inclination to use this understanding in flexible ways to make mathematical judgements and to develop useful strategies for handling numbers and operations.

Addition

The answer is attained by adding the tens first, then followed by the units of each segment.

28 + 13 = 41

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The Number Line

Young children could draw a number line and fill in the required numbers for addition or even subtraction.

Addition operation in number line

27 + 28 = 55

Subtraction operation in number line

55 – 28 = 27

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Hundreds Chart

There are two type of this chart, 0 – 99 and 1 – 100. The 0-99 chart is usually used for subtraction while the other one is suitable to be used for addition.

0-99 chart

1-100 chart

How to use it?

Based on the 0-99 chart, the block is placed on the number 52. Question given by the teacher is 52 – 19.

Young children know that by moving the block 1 move upright is equal to 10 moves to the left which is equals to 52 – 10. By now they would only need to move 9 move to the left (for subtraction) to answer the question. The picture below further explained on how the operation is carried out.

Workshop 1

Beginning Processes

Irons (1999) asserts that young children will start to learn mathematics when they begin to interact with the world around them. They are also become aware of the attributes of everything that their eyes could see, their hands could touch, their ears to could, their nose could smell and their tongue could taste.

Attributes are features or characteristics of a certain object which young children could notice through the colour, mass, shape or even sizes as claims by Irons (1999). The process of identifying and describing attributes is concerned with the likenesses and differences between objects or pictures (Irons, 1999). The ability to recognize likenesses and differences is important in the development of children's beginning mathematics (Irons, 1999). The designed activities should be able to help young children to distinguish details and be more attentive. There are namely 5 beginning processes:

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Matching

Matching activities focus on the ‘sameness’ of attributes (Irons,1999). Children will select the objects or pictures that have the same attributes and group them together. They will then describe why the objects or pictures match using the language associated with the attribute that they have selected. For example they could match the objects according to the colour, shapes, functions or even the size of the objects or pictures.

The picture shows several toy vehicles with varities of colours were laid on the table. Children were asked to match them according to their attributes. They have matched those toy vehicles into their types. Similiarly, young children could match those toys into other attributes such as to match them according to colour or even match them to according to the shape of the toy vehicles as long as there are ‘sameness’ among them.

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Sorting

Irons (1999) claims that sorting involves large number of objects and the process of sorting involves grouping objects or pictures according to more attributes. Children are the one who should choose the attributes to work with (Irons, 1999).

In the picture, children use the sorting board to sort the shape according to the colour and type of the shapes.

The process of sorting involves the three 'Ds' - Decide, Do and Describe. The aim is to have children decide which attributes to sort by, do the sorting and describe the sorting after it has been completed (Irons,1999).

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Comparing

Comparing is based on relations between amounts of an attribute that two objects possess. Children are required to determine which of two objects possesses more or less of the attribute (Irons, 1999).

Children could see the difference between these two objects in term of their size. The one on the right is bigger than the one on the left. According to Irons (1999), children’s vocabulary is greatly developed during the process of comparing due to the uses of phrases such as ‘more big’ because it makes sense to them. Teacher should model the correct usage of words that associate with comparisons of objects such as the ‘er’. For example, the object on the left is big but the object on the right is bigger.

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Ordering

The process of ordering according to Irons (1999) is the involvement of arranging objects according to the relations between them based on the increasing or decreasing amounts. Either from big to small vice versa.

Based on the picture above we could see that young children are able to arrange the order of the block in an increasing order based on the size from smallest to the biggest.

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Patterning

Patterns are formed by the repetition of objects or pictures and are recognisable and predictable in order of any arrangement (Irons, 1999). There are three type of patterns; repeating patterns, growing patterns and relationship patterns. Repeating patterns are usually used for young children because it involves only one element that is being repeated.

Repeating Pattern

Growing Pattern

Relationship Pattern

Number of boxes	Number of crayons
1	8
2	16
3	24
4	32