Thursday, September 30, 2010

I have 10 cookies ^_^

For record, I have achieved 10 cookies for my participation in class. Encouragement "in kind" works to promote learning too! Thank You!

Wednesday, September 29, 2010

Reflection on Elementary Math

"Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. (NCTM, 2000, p.20)

The learning principle summarizes the six lessons that I had in class. In my years of learning, Mathematics was my favourite subject because it was easy to get good grades with lots of practices; not because I understood the concepts behind it.

I looked forward to attend every class; as I have learned and understood Mathematics in a different light. This time, I know why I am learning Mathematics, as I was LEARNING BY DOING! The practical knowledge in class, coupled with the knowledge acquired from the textbook, prompted me to reflect on my teaching practices.

In the preschool years, we followed Jerome Bruner's Concrete-Pictorial-Abstract (CPA) approach for teaching of concepts. We have been teaching the concepts according to the curriculum planned, but, did we manage to 'convince' the children that learning Mathematics is fun and magical?

Children, as well as teachers, learn by inqury, communicating with their friends, reflecting, doing, and collaborating. As teachers, it is our responsibilty to provide appropriate learning materials when we are teaching new concepts, and importantly, we must design activities with variations to strengthen children's learning. Zoltan P. Dienes's Mathematical Variability Principle suggests that the "generalization of a mathematical concept is enhanced when variablesw irrelevant to that concept are systematically varied while keeping the relevant variables constant" (Post, 1988).

The activities in the class all geared toward getting us to understand number sense, visualisation and patterning! Through the activities, I am more careful with details; trying to find a relationship or pattern among numbers.

My school will be celebrating Children's Day tomorrow, and I will be performing magic using "spelling number" and "dice trick" learned in lesson 1. I want to excite the children and arouse their curiosity minds, in all my future Mathematics classes!

And yes... we must be meticulous in the Mathematical terms that we used in class. Similar is DEFINITELY not the same as identical!

Post, T. (1988). Some notes on the nature of mathematics learning. In T. Post (Ed.), Teaching Mathematics in Grades K-8: Research Based Methods (pp. 1-19). Boston: Allyn & Bacon. Retrieved September 29, 2010, from


Pentagon - polygons with five sides. In the class, my first reaction to the total interior angles is 360 degrees. However, after folding the paper into numerous squares; I realised that the sum of interior angles is 540 degrees, with four right angles and two 45-degrees angles!

The development of geometric thinking by van Hiele enables us to understand the five levels of thinking processes used in geometric contexts. For the six year olds children, I believe they would have reached level 1: analysis of the van Hiele levels of thinking processes, with some children achieving level 2: informal deduction where children are able to understand the various properties of shapes. Understanding the children's level of development enable us to plan meaningful developmentally appropriate activities for the children.

The "Shape Sorts" activities in the textbook facilitate learning of mathematical concepts, and promote language skills and logical thinking skills. As educators, we must give time for "students to do activity using ideas they own and understand" (Van de Walle, 2010, p. 406).

In the photograph, the children decided to share a piece of square bread. So, they requested the teacher to cut one square bread into two triangles; they could have asked the teacher to help them cut into two rectangles. The simple activity that the 3 year olds children engaged in showed that they have learned geometrical concept, developed spatial visualisation skill, appreciation of friends as well as capable of using ideas that they have learned earlier in class.

In class, we used tangrams to form a square. This activity was exciting as we had to race against time to form square with increasing pieces of tangrams used. Though there are only 7 tangram pieces, the learning opportunities are unlimited. Through playing with those tangrams, we realised that two small triangles make a paralleogram, a square and even another bigger triangle. It can also be used to build different objects with odd shapes or use it as an estimating tool.

In the class, I learned that 'diamond' is not a shape. I may, in my years of teaching, been teaching diamond as one of the shapes. This lesson prompted me to research more on shapes and the website link below reaffirmed that diamond is NOT a shape!

It is important, for educators, to research and refine our knowledge, prior to teaching concepts and knowledge to children.

Website link for shapes:

Teaching of Number Sense

I agree with the Van de Walle (2010, p. 125) that children come to school with their ideas about number and our task is to help them develop "new relationships" and extend their understanding.

The framework of the Mathematics curriculum by Ministry of Education (MOE, 2006) states the learning of mathematics involves more than learning of concepts; it involves the understanding of underlying mathematical thinking, the strategies to problem solve, the positive attitudes, and the appreciation of mathematics as an important and powerful tool in everyday life. This is congruent to the big ideas by Van de Walle (2010, p.125) when he states that the "number concepts are intimately tied to the world around us" and by applying number relationships to the real world settings, it marks the "beginning of making sense of the world".

In the preschool, the number concepts that we introduced are:
1) the relationships of more, less, and same
This is one concept that we can introduce to the preschool children. Children as young as 3 year olds are able to tell that a tray with five bear counters is more than a tray that contains 1 bear counter.

2) Problem structures
Be it join problems, separate problems, part-part-whole problems or compare problems, our school curriculum only covers the simplier problem structures that require less than 3 steps to solve the questions given.

In the preschool years, children learn from concrete to pictorial to abstract (CPA approach by Jerome Bruner). When doing the problem sums, we use concrete materials (e.g. unifix cubes or counters) to help children understand and reinforce learning through the introduction of models. We did not use the number line concept to solve story problems. I believe, both methods work equally well, as long as we introduce the appropriate tools to the appropriate age group to help them understand what is happening in the story problem.

Other mathematical concepts that we taught in the preschool are:

1) Numeral writing and recognition

2) Counting on and counting back

3) Patterned set recognition

4) Part-part-whole relationships (based on single number)

5) Addition and subtraction

6) Estimation and Measurement

7) Data Collection and Analysis

In my school, we did not teach the following concepts:
a) Doubles and near-doubles
This is an interesting concept and I would like to try this out. When children understand part-part-whole concept, the doubles and near-doubles strategies strengthen children's understanding of basic addition facts. However, this can be challenging, even for the K2 children.

b) Anchoring numbers to 5 and 10
Though teaching the anchoring numbers to 5 and 10 is an useful way to help develop thinking about various combinations of numbers, it is not an easy task for the 6 year olds children. The five-frame or ten-frame provide opportunities to train children to "see" the relationships among numbers, and when the children are familiar with the relationships among numbers, we have also trained them in terms of mental computation skills.

The development of number concepts and number sense cannot be underestimated, and children need to have a good foundational ideas in order to extend their learning to larger numbers and computation.

Using Technology to Teach Mathematics

In the Ministry of Education (MOE) website, one of the aims of Mathematics education is to enable the students to "make effective use of a variety of mathematical tools (including information and communication technology tools) in the learning and application of mathematics" (MOE, 2006).

Time has changed. I was allowed to use calculators in Secondary School. After three decades, students are using calculators from Primary 5. My daughter was one of the first batch of students that were required to use calculators in their Math lessons, perhaps one day, the Primary 1 students will start to use calculators in their studies too.

Reading Chapter 7 changed my perspective on the use of technological tools to guide our students in their learning. In the preschool, we make use of computers, interactive white board, computer software, and projectors to facilitate children's learning. It did not occur to me that calculator is one of the tools we can utilize to teach Math concepts.

A simple "0+1===" enables the children to "count by ones" on the calculator. We can make use of "o+2===" to teach counting by twos, "0+3===" to teach counting by threes and so on. Yes, this makes counting easy and fun for the preschoolers, and we can place the calculators at the Math learning centre too. However, is the usage of calculators appropriate for the 2 to 5 year olds, or even the 6 year olds?

According to Jerome Bruner's Concrete-Pictorial-Abstract approach, children learn through concrete materials such as unifix cubes, counters, base-ten blocks, and objects that can be physically manipulated. Children must reach the abstract level in order to understand symbolic representations such as numbers. At 4 year old, children may not understand the concept of "0+1". To further complicate matters, there are 3 equals after "0+1". It is not easy even for a 6 year olds child to understand why we need to press 3 "=" after "0+1" to "count by ones" and children may assimilate the same in their daily work too. The challenge, then, falls back at the educators.

As educators, how can we get the children to differentiate the language used in their daily work is different from "technology language"? Is it too early to use the calculator in the preschool years?

I agree with the author that calculators can be used for "exploring patterns, conducting investigations, testing conjectures, and solving problems", and should not be used for "practicing computational skills" (Van de Walle, 2010, p. 112). So, is using calculator age-appropriate in the preschool? In my opinion, use it sparingly.

I like the website "illuminations" link from the blog. It is a comprehensive and easy to use website and the online activities are fantastic for all ages. I especially like the "Bobbie Bear" online activity. In this activity, the children are supposed to guess the number of outfits that Bobbie Bear can mix and match. Children first make a guess, and then try out the various options of mixing and matching Bobbie Bear's outfit. When all options are used up, the door will be closed and Bobbie Bear can embark on his vacation.
Indeed, technology tools provide many avenues for children to engage in learning. The sound effect, colours, animation, and varieties of programmes available naturally lure our interest in learning. A 2 year olds girl in my school uses her mother's iphone4 to read e-books, play games and watch cartoons; she is one of the few children in the Playgroup class that could recognise numbers, shapes, and colours. Technology enables the children to learn in a different light, however, we must not forget the wonderful moments of learning through the "traditional" methods; learning through play, teamwork, hands-on activities and many others.

Ministry of Education (2006, March). Mathematics Syllabus Primary. Retrieved September 29, 2010, from

Van de Walle, J. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). New York: Longman.

Friday, September 24, 2010

Sequencing Learning Tasks for Place Value

Everyday, there are many opportunities for children to practice counting. Without prompting, my N2 children will "report" to me the number of children that are present and absent for the day. With half-day programme children leaving the school after lunch, the children will count again and tell me the number of mattresses that I have to prepare for them during nap time.

All of them know how to count beyond fifty; some even beyond hundred. However, after reading Chapter 11 and attending the class on place value; it made me wonder if my N2 children comprehend the whole-number place-value concept.

Few days ago, I did an experiment with the N2 children. I placed fifteen unifix cubes on the table and delibrately connect ten unifix cubes in a row, followed by placing the balance five unifix cubes loosely on the other end. When I asked the children to tell me the number of unifix cubes I had, more than 90% of the children counted every single one of the unifix cubes. One girl, who was observing her friends, started counting at 10, then 11, 12... till she reached 15. I was surprised with her strategy and asked her why she started counting from 10. Her answer? She said many friends counted "10" for the row of unifix cubes, so it must be 10 and there is no need for her to count again. What a smart observation!

"Children count out the tiles one at a time and put them into the pile with no use of any type of grouping" (Van de Walle, 2010, p. 188). "Counting plays an important role to scaffold children's construction of base-ten ideas" (Van de Walle, 2010, p. 189). The N2 children are great at counting, one at a time, and the simple activity enabled me to understand their present level of development.

I agree with Jerome Bruner's instructional approach to learning mathematics, where teaching of concepts should be based on Concrete-Pictorial-Abstract (CPA) approach.

With the CPA approach in mind, I hope the sequencing of learning tasks for place value put forward will scaffold the children's understanding of the tens and ones concepts better.

My sequence of learning tasks for Place Value is as follows:
1) Place Value Chart
From the earlier lessons on three bundles of ten and four sticks, 3 groups of ten unifix cubes and four unifix cubes, and the introduction of dime and cents, the children will have a good understanding of 3 tens and 4 ones. Thus, to place the number 3 in the 'tens' pocket and 4 in the 'ones' pocket is congruent to what they have seen in the earlier lesson.

2) Tens and Ones notation
From the Place Value Chart, children will have a good idea that the 3 is in the tens place and 4 is in the ones place. As such, to introduce the 3 tens 4 ones at this junction is an extension to the earlier concept.

3) Number in Numerals
Next, I will introduce the number "34" to the class. This is simply a transfer of concepts learnt from the Place Value Chart and "tens and ones notation" to the numeral "34".

4) Expanded Notation
By now, the children will have a good understanding that 34 is make up of 3 tens and 4 ones. To reinforce children's learning, I can further expand 3 tens to 30 and 4 ones as 4. This acts as a check on whether the children have fully comprehend the place value concept.

5) Number in Words
From part 4, children know that 34 is make up of 30 and 4. To teach thirty-four at this junction is appropriate as children literally say thirty-four from the expanded notation of 30 - 4.

The objective of the whole-number place-value concept is to get the children to understand the base-ten ideas. With learning activities that integrate the "grouping-by-tens", the ultimate aim is to get the children to learn "through reflective thought" (Van de Walle, 2010, p. 189).

Reference :
Van de Walle, J. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). New York: Longman.

Friday, September 17, 2010

Problem Solving in relation to Environment-based Task

Chapter 3 and 4 of the textbook provide me a good insight of why we should TEACH THROUGH PROBLEM SOLVING.

"Good problems will integrate multiple topics and will involve significant mathematics" (Van De Walle, Karp, & Bay-Williams, 2009, p.32). That set me thinking of the challenges to plan a meaningful lesson for my children. Am I not doing the right thing so far?

I found my answer from the phrase, "children are learning mathematics by doing mathematics!" (Van De Walle et al, 2009, p.33). When we plan our lessons, we always plan according to the children's level of development, and most importantly, what they are interested to learn.

I agree that teachers require a paradigm shift when we want to teach through problem solving. In fact, I have to learn more ways on how to engage the children to learn content eagerly. On the other hand, it is imperative to prepare learning materials and select appropriate manipulatives if I want the children "to learn content by figuring out their own strategies and solutions" (Van De Walle et al, 2009, p.34).

I am teaching the Nursery 2 children this year; a fantastic group of children! Each of them has a different learning style and different level of development (and they should be!). Now, it is time to rethink and redesign my teaching methods. A major shakeup time for me!

I had a good understanding of chapter 3 and 4 contents when we were carrying out the environment-based task. It is not just trying to find learning contents to teach from our environment or find any environment and try to fit it to our learning contents that we deemed fit.

After reading the textbook, I realised the Three-Phase lesson format is also applicable to our environment-based task.

Firstly, the BEFORE phrase of a lesson must be clearly established. From our daily experiences with the children, we need to understand what interests them, what are their prior knowledge and experiences, and are they ready to problem solve with me?

During the DURING phrase, we are just like the students, excited and ready to GO and tried to find the environment suitable for our learning contents. In the AFTER phase, we SHARED ideas and reflected on what we have created for the children.

The age group that we are targeting for our environment-based task was the Kindergarten 2; as the content initially planned for was numbers and operations - to learn addition and subtraction. The objective of using the nine "squares" on the metal gate was to get children to fill up the spaces using either addition or subtraction of whole numbers. In fact, we had fun trying to outwit each other to complete the nine squares.

During the AFTER phase, we reflected and realised that the contents can be modified to "make the task accessible to all students" (Van De Walle, 2009, p.65). In this case, if we want to teach the Nursery 2 class, we can ask the children to complete the nine squares by filling unrepeated numbers from 1 to 9. In fact, we can scaffold their learning by increasing the number of squares, as long as the children are having fun learning.

During our reflection stage, we realised that the metal gate is a fantastic learning tool for charts and graphs learning too. Thus, learning is multi-leveled and we can integrate multiple topics for this simple learning material.