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Mathematics and Early Childhod Educator

This course has helped me to view Mathematics in a wider perspective- Mathematics is Problem Solving. Knowing and understanding the curriculum framework of Mathematics in Primary School, I am aware of the importance of building solid foundation in Preschool Education.

How do I apply that awareness in my teaching practises to young children?

• introducing children proper Mathematical Language, such as, take away.
• use clear, complete and 'make sense' language when explaining, teaching, or giving instructions- for example; when teaching about the concept of Ordinal (position) number of 1st, 2nd, and 3rd, the teacher asks (after drawing on the board), "Children, can you tell me which ball will reach to the finishing line first? Is it A, B, or, C?". -- What do you think of the question posed to the children? Do you think it is appropriate? (A ball is not able to roll by itself, without someone kicking or rolling it, and this has to be conveyed to the children). Therefore, using balls as an example to teach Ordinal Number concept is not appropriate.

I strongly agree with what our dearest friend said, Jerome Burner, that children should be brought into three stages in teaching Mathematics; Concrete- Pictorial- Abstract. Therefore, as an early childhood educator, it is necesseary to provide developmentally-age appropriate concrete materials for children.

• give opportunities to children to build on critical thinking skill by providing concrete materials
• teaching Mathematical concepts through Literature (story books)- Pictorial

This module has been an eye-opener for me, especially in understanding how Mathematical concepts can be taught to young children.

Here is an article that talks about the connection of Math and young children;

http://www.walearning.com/articles/math-language-and-young-children/

I am currently working on my Action Research assignment, which topic is on evaluating Math learning corner set-up in enhancing children’s independent learning using a lesson study. I am gathering all my research data and materials at the moment, and looking forward to apply all those knowledge which I gained from this Math module, and put them into practises, for this assignment especially.

Throughout this course, I have collected.....................my one and only dearest ONE cookie, =D or =( ??

In search of the Polygon's Interior angles

On the fifth lesson, I was given a piece of square-shape paper.

Dr.Yeap: "Class, please turn this square-shape paper into a five-sided Polygon; you can cut, fold, or tear or  anything to turn it into a five-sided Polygon".

I questioned myself, what is Polygon? The last time this word came to me was few years back on my E-math 'O' Level paper". Well, does not matter, since I understand what does 'five-sided' means. Then, I folded oe side of the paper into a triangle. It looks like this:

                                                                Fig.1.Five-sided Polygon

Dr.Yeap:"Please find all the sum of the interior angles"

I am rather confident that the sum of those three pependicular angles at the corner are equivalent of 270 degree. But, how do I find the other two angles which are at the bottom? I was not so sure, but somehow I perceived that there has to be relation with the properties of angles in triangles in order to find this interior angles. 

Through observation, my thinking was reaffirmed that this polygon is a combination of a right-angle triangle and a trapezium. 

Here is my solution:

                                           Fig.2. Solution of Finding the sum of Interior Angles

After doing this exercise, I read the chapter on Geometry and my understanding got deepened; the concepts of right, obtuse, and accute angles, congruence of line segments and angles, and symmetry is a good way to help us to see how different collections of properties apply to special classes of shapes.

Then I went on to do my searching on the term Polygon; is a closed plane figure with three or more, usually straight, sides. Ehm,..three-sided? Does it mean triangle is considered as a Polygon as well? Yes, it does. 

Triangle is considered as a Polygon! A ha, now I remembered it! It was once said by my Math teacher back in Secondary School.

Whole-Number Place-Value (Number Sense) Concepts in Preschool

According to the MOE curriculum guidelines, for Numeracy skill, Preschool Education is to provide opportunities for children to:

• Understand the concept of one-to-one correspondence
• Understand and use numbers in daily experiences
• Understand the ordinal, nominal, and cardinal aspects of numbers (1-30)
• Understand the concept of additon as combining two groups of objects in small quantities
• Understand the concept of subtraction as taking objects away and find out how many are left or comparing groups of objects to find out the difference between them. 

There is one point in this particular chapter mentioned which brought me upon a realisation, that, children may count "ten, eleven, twelve,....twenty" and so on without realizing the "twenty-ness" of the quantity.  It reminds me to some of my students, whom are able to do rote counting from 1-20 pretty well, but when I asked them further to show me the quantity of such numbers (by giving concrete objects), some of them were not able to do so. They made a few attempts by counting from different direction; beginning, middle, or end. Eventually, they still did not manage to give the quantity accordingly.

After reading on this chapter, I began to see the importance to connect the count by-tens method with their understood method of counting by ones, the children need to count both ways and discuss why they get the same results. Therefore, developing base ten concepts of lesser quantities (perhaps up to 20) is necessary to be included in the Preschool curriculum. 

The role of counting should be further broken down into three stages; counting by ones, counting by groups and singles, and counting by tens and ones. It is insufficient for teachers to tell children that the counts will results for the same quantities. The three stages in counting is important for students, as it is the relationship they must construct themselves through reflective thought. 

I am currently teaching a group of Kindergarten One level. Here are some the objectives of Mathematics Lesson Plan for K1 level: 

• rote counting 1-20 or more
• count realiably, one object at a time up to a collection of 10 or more
• know and show that the quantity of a set up to 10 or more is the "same" irrespective of strating point of counting
• matc up to 10 or more: number names with quantity, number names with numerals, numerals with quantity, number words with quantity, number words with numerals
• use "one more" and "one less" for quantities up to 10 or more
• recognise that the quantity of a set of objects is the same when arranged differently

Those objectives are good foundation in building a number sense, in which I believe would create a smoother transition in teaching whole number place value concepts. 

The three stages of counting; counting by ones, counting by groups and singles, and counting by tens and ones should be further emphasized to teach concept of Numbers, for the Preschool Curriculum Framework.

Technology Can Improve Attitudes and Motivations

Initially I was not convinced that calculator is beneficial for younger children because in my opinion, it can bypass the leaning process which often more important than simply knowing the final result. This can hinder the development of their intuitive and rational thinking. For example, student might know that 5x4 is 20 by pressing the buttons, but would they know that 5x4 is equivalent to the process of adding 5 for 4 times? 

But interestingly Chapter 7 on the use of Technology to teach Mathematics presents an approach where we can utilize calculator for the better education. Research shows that calculator can boost student’s enthusiasm to learn mathematics (Ellington, 2003). The presence of calculator will provide them accurate answer, therefore it lessen their anxiety of not getting the correct answer in the end. 

I chose a Math-activity taken from this website; http://nlvm.usu.edu/en/nav/topic_t_1.html , on the topic of ‘Place Value Number Line’ to strengthen my point of the positive used of technology in mathematics learning.  

This activity would enhance students’ enthusiasm through the visualization of mathematics. For example, the screen-capture below (fig.1) allows the users to have better sense of number with relative to other numbers (e.g. number 580 is positioned about half of the scale range from 0-1000). Now student would find math more interesting as they can use their sense to relate with mathematics problem.

Fig.1. Number within hundred intervals

From fig.1, at the bottom of the screen, there are three options available: explore, practice, and test which the difficult level increases accordingly.  In the ‘test’ mode, the application does not allow students to undo their steps, whereas in the ‘practice’ mode, students are given opportunities to practice their understanding and skill, the "undo" function can be applied. In this mode, the history of children steps are tracked, so that children are able to track down on what they have done. 

In the ‘explore’ mode, students are given opportunities to do Mathematical exercises on Number Line aided by this 'explore' mode. 

Fig.2. Different scaling in number 

From fig.2 , the top of the blue triangle indicates the position of the chosen number within the hundred scale line. This will help to visualize the location of the chosen number within the hundred scales. In addition students can use the zoom in and zoom out function. Zoom in will allow users to have a more accurate visualization of the location of a specific number on a number line by bringing the scale of the numbers down (e.g.from hundreds to tens). 

Surely we understand better when we see!

Teaching through Problem Solving

Teaching through problem solving accomodate to the diversity of different learners; as in problem solving, students are given opportunities to do the "talking" and teacher would only share relevant informations, as long as the tasks still remain problematic for the students.

Problem solving allows children to "make sense" of their own understanding through thinking. I strongly believe, when students reach this stage of understanding, they will have a solid understanding on the concepts taught. 

Myself and group members have chosen the topic on Pictograph, to introduce a new method of collected data or information into a Pictograph. The sculpture which is located at the Singapore Art Museum is the targetted object.

Fig.1. The Sculpture

 This activity requires the children to do a shape hunting activity in which they are to identify what types of shapes are there on the sculpture. After identifying the types of shapes, they have to count how many shapes are there for each type. The challenge for this task is to count the number of spheres. If you take a closer look on the sculpture picture above, there is one sphere which most of the part is covered by other spheres; therefore, it is a little difficult to identify that particular sphere. Furthermore the spheres are arranged in circular motion. 

Polya described four steps in problem-solving process:

1. Understanding the problem

2. Devising a plan

3. Carrying out the plan

4. Looking back

In the Pictograph task above, the students will not be told strategies on counting the spheres. However, they will be given instructions (only if they are not able to spot the hidden sphere after making a few attempts),such as, "Please take a closer look on the sculpture, can you spot whether there is another sphere inside.?"
Therefore, students will be able to identify the problems of the task on their own. Once they are able to derive to the first step on their own, they will be able to proceed to the next three steps with teacher as facilitator.

Sequencing Learning Tasks for Place Values.

In referring to teaching variations to children, place value chart should be introduced right after teaching the units using sticks (concrete materials). It is because, the sticks itself are in a bundle of tens and ones, which are clearly stated in the place value chart above the numbers. Therefore, it would guide students to a smoother transition in thinking and they will be able to comprehend that 30 is make up of 3 tens and 4 is make up of 4 ones; which explains the reasons why there are 3 bundles of sticks and 4 loose sticks. 

The numeral words in the expanded notation do not indicate anything. It simply showing the number 30 and 4, followed by the words.There is no bridge between the expanded notation and the sticks that will aid students to connect their understanding of 3 tens and 4 ones make up to the value of 34. The expanded notation is more abstract for the children, which ideally come right after the place value chart.