Session 6 [27 August 2011] - Measurement

In kindergarten, our children learn about measurement by using non-standard units such as cubes, sticks, pencils….etc. Our teachers also used simple story sums to ask children to measure the length of the whiteboard or the height of their chairs. but I think storytelling is also another interesting way to stir up children’s curiosity in learning about measurement.

Today, Dr Yeap told us an interesting story titled “How Big Is A Foot”.

As the story was unfolding, I found myself wondering how the queen’s bed would be measured. I enjoyed how the story taught us about problem solving. We can see that through problem solving, children learn to explore, think through an issue and reason logically to solve a problem.

As for the outdoor activity on “measuring the height from the ground level to the basement of the MRT station”, our group started to count the number of stairs(16) per level and multipled it by 4 levels, to get the total number of steps. To reconfirm the total number of steps, we actually went down and counted one by one. However our initial conclusion was wrong; the correct answer is 62 steps. We also measured the height of the steps. Our findings were:

841 + 54 = 895cm

The height is about 895cm

Next, we were asked to make a box that could fill up to 15 kidney beans. With 1 bean in hand, we estimated an approximate amount and came out with a small box. Amazingly, this small box could fill the 15 beans.

Activities such as measuring of the height of the MRT station to the making of the small box aided us in our learning on estimating measurement. Measurement is an important way for young children to look for relationships in the real world. I believe that teachers need to provide many opportunities for children to practise measurement in order for them to be able to figure out how big or little things are, in addition to problem solving.

Session 5 [26 August 2011] - Geometry

In class, we joined dots to form squares, triangles, rectangles… polygons. What are the interesting things about these shapes? We learnt that if a shape appearance is changed, it is called transformation and it can be done in many ways including :

- Translation

Translation -Move up or down, side to side but do not change its appearance

- Rotation

Rotation – figure rotated about a point

- Reflection

Reflection – figure reflected over a line

In preschool, children manipulate with a variety of both 2 and 3-D shapes. They do sorting and classifying, trying to see how shapes are alike and different. They build, put the shapes together, take them apart and even draw the shapes. In the process of manipulating with shapes, children are developing an understanding of geometric properties.

The Van Hieles theory of geometric thought features the 5-level hierarchy of ways of understanding spatial ideas:

0. Visualization – groupings of shapes that seem to be “alike”

1. Analysis- classes of shapes rather than individual shapes

2. Informal deduction – the properties of shapes

3. Deduction – relationships among properties of geometric objects

4. Rigor – deductive axiomatic systems for geometry

To help children to grow and develop the ability to think and reason in geometric contexts, teachers have to provide many opportunities and rich experiences for children to build up an understanding of the systems of relationships between geometric ideas.

Session 4 [25 August 2011] - Fractions

1. Let’s divide the rectangle into 4 equal parts.

How many ¼ (one fourths) are there?

1 divide ¼ = 4

- The top number counts(numerator)

- The bottom number tells what is being counted(denominator)

2. Let’s divide the rectangle into 3 equal parts

1/3 is green

3/9 is green

Now both have the same amount of green

So we say 1/3 is equal to 3/9 or 1/3 = 3/9

3. What is one third, one fourth?

They are fractions. A fraction describes a part of a whole when the whole is cut into equal parts.

1 Whole

2 Halves

4 Fourths

8 Eighths

Students must come to understand that a fraction does not say anything about the size of the whole or the size of the parts. A fraction tells us only about the relationship between the part and the whole.

When partitioning a whole, students need to be aware of 3 aspects of fractional parts:

1. The number of parts determines the fractional amount

2. The parts must be the same size, though not necessarily the same shape

3. The number of parts that make up a whole determines the name of the fractional parts

In order for the children, to grasp the concept that fraction segments are part of a larger whole, we need to continually expose children to various games and experiences that are familiar to them. For example,  under our preschool context we usually use pretend food items to teach the children simple concepts of fractions (i.e. half, 1/4, 1/3 ). This will enable them to visualise fractions and understand that they are part of an object. They can gradually move on to the basics of adding and subtracting with fractions with the use of the teaching aids. We can expose the children to writing fractions but it might be a little too abstract for preschool children to comprehend.

Session 3 [24 August 2011] - Lesson Study

The class was asked, "What is Lesson Study?"

Lesson Study is a new approach to me. My understanding is that a small group of teachers work collaboratively to plan, teach, observe, analyze and then refine the class lesson. The video showed today on lesson study conducted by a preschool teacher proved to be insightful. Preschool teachers can conduct a lesson study:

1. To improve their teaching
2. To better understand how children learn
3. To build a pedagogical knowledge of teaching

When conducting a mathematical investigation, it is important that teachers plan an activity that is divergent in nature so as to allow children to explore and experiment mathematical ideas and situations in many directions. At the same time, allowing children to demonstrate problem solving, thinking and communication skills.

In mathematical investigation, teachers:

- Need to observe
- Listen to follow children’s reasoning
- Ask questions at the right moment to help children think further
- Provide waiting time for children to think and discuss ideas
- Act as a facilitator

Certain aspects of it are already in practice I believe, as preschool teachers need to observe and facilitate their children on a daily basis.

As such, lesson study provides opportunities for teachers to reflect and research on ways to improve their teaching practices. It is also a process where teachers can assess their children based on their ability to describe and communicate their ideas, how they justify solution and the steps they have taken to arrive at their solutions.

Session 2 [23 August 2011] - Reasoning and Making Sense

Today, I learnt a new word “subitize”. It is a way of manipulating numbers through visualization. When we look at the things, we do not count them individually but in groups. We recognize and know the amount. For example, when we roll the dice in game and we know what the number is without counting each dot on the dice, we are subitizing.

Another example is the game “Pick up Sticks” we played in class. It’s a game for 2 players and they have to take turns to take away 1 or 2 sticks from her partner. The winner is the one with 1 or 2 sticks. In the process of playing, children learn :

1. To count – how many sticks
2. To look for patterns – number sequence
3. To problem solve – how many stick do I need to remove to be the winner
4. To subitize – visualize the things and know how many

Other games such as Dominos or games that involve the rolling of dice will also help the children to visualize the numbers on the dots and gradually to have a better understanding of mathematics and its relationships.

The dice game that Dr Yeap conducted with the Primary 1 children stirred up my interest in finding the ‘hidden’ answer. Using 2 dice, and with each end joined together, he asked the children to guess the sum of the 2 hidden numbers. The children were asked to reason, to look for pattern and to communicate their ideas to find the answer to the question. Similarly, when children are face with word problem, they are at a loss of what to do. Teachers need to guide them by providing opportunities for them to:

1. Talk about what the answer might look like
2. Think about the problem and what it is about
3. Think of the answer before solving the problem

In the process, teacher has to ask the children to focus on the problem and the meaning of the answer, and to focus on the structure of the problem. Eventually, the thinking will lead to a rough estimate of the answer. The most important approach to solving any contextual problem is to analyze it and make sense of it. We can see that at the end, mathematics is about reasoning and making sense of situations.

Session 1 [22 August 2011] - Number Sense

A teacher of mine once told me “this addition worksheet is too difficult. My children cannot count”. At that point in time, I wonder why the children cannot count. I thought perhaps the children have not grasped the concepts of counting or they have not got the awareness of the meaning behind the numbers.

As I reflect on today’s lesson, I learnt that there are 4 pre-requisite to counting:

1. Classify
2. Rote counting
3. One to one correspondence
4. Appreciate last number counted


These 4 pre-requisite are the foundations of early mathematical concepts. Hence it is important that teachers provide opportunities and concrete materials for children to experience hands-on activities that will help to develop these skills and enhance school readiness.

One important model that I have learnt is the ten-frame model which allows children to think of number in relation to 10. For example, provide children with about 10 counters. Have them place 6 counters on the ten-frame. Teacher may ask: How can you make 6 on the ten-frame? Can you show me a different way to make 6 on the ten-frame? I believe this activity will help children to make representations for numbers 1 to 10, practice counting by adding on or taking away  the required number. How the children are using the ten-frame provides teachers with insights into their number concepts development.

As teachers, we need to help children to understand the different uses for numbers, to help them to develop multiple ways of thinking about and representing numbers so that they will have the ability to count accurately, count on from a specific number as well as count back, and to see relationships between numbers. Gradually, children will develop number sense as they continue “to explore numbers, visualizing them in a variety of context and relating them in ways that are not limited by traditional algorithms”(Howden 1989).

Learning Mathematics the fun way!

           When I was in Primary School, I enjoyed doing simple Mathematics.  As I entered Secondary School, Mathematics was not an easy subject for me to learn. It was no more simple addition and subtraction. I had difficulty trying to understand the concepts being taught. The teacher was just teaching and ‘drilling’ the class. What she wanted to see was the product. I could not catch up with her teaching and felt left out. I did not do well.  


Learning Mathematics through counting



When I got hold of the book “Elementary & Middle School Mathematics”, I thought “Oh no! This book is not for me. I am no good in Mathematics.” Anyway, I still have to read to seek new knowledge, and to find solutions in helping children to enjoy doing Mathematics. I believe it is the teacher who will shape mathematics for the children they teach. Thus, “learning mathematics is maximized when teachers focus on mathematical thinking and reasoning” (www.nctm.org).

                As I turned to Chapter one on page 2, my attention was caught on the Principles and Standards for School Mathematics recommended by NCTM. I strongly agree that school should follow the guidelines of the Six Principles, especially the "Teaching” and “Learning”. I believe children will learn and have the ability to think and reason mathematically in order to solve problem when teachers are equipped with the appropriate instructional tasks and strategies that will enhance students’ learning. Besides teachers also need to understand how children learn mathematics.

Child working on patterning

I think the use of computer and calculator in kindergarten will stifle the children’s learning and creativity as they are seen “talking to a programmed machine.”  I want children to interact with concrete materials, solve problems with teachers and peers,  evaluate their work and those of their friends and to share ideas with each other as these will enhance their problem solving skills.

I believe teachers should provide opportunities for children to build connections between what they know and what they are learning.

What strikes me most is The Five Process Standards on page 4. Though I am not teaching a class, I want to share with my teachers the importance of providing opportunities for children to experience the processes of problem solving, communication, reasoning, connection and representation. I believe children are by nature inquisitive and are eager to learn new things. Teachers should let children explore and think of many ways to solve different kinds of problems, such as through the use of manipulatives  where children learn relationship between groups of things to be sorted counted, shared and represented and also through interacting with words, pictures and symbols to clarify thoughts.



Making connections through play (i.e. one to one correspondence - 6 seats to 6 sets of cutleries)

"I can count!"

Most importantly, I believe teachers have to be equipped with the appropriate mathematical approach instruction so that they can help children come to make sense of Mathematics while enjoying it.

Ultimately, teachers “need to have a profound, flexible, and adaptive knowledge of mathematics content” (Ma, 1999). Teachers also need to have persistence, positive attitude, readiness for change and reflective disposition, which are equally important to succeed as a teacher of mathematics.