Sunday, 7 April 2013

Graphing & Final Say

We have come to the end of the session in learning Mathematics. I'm very much interested in the the concept on graph. I learnt that graphing has a purpose in doing as well.

Purpose of graphing

  • How to categorize ?
  • Who is it useful for?
  • Who would need to know such information or data?
As a class, we listed down a few possible categories to create the graph. Such examples would be how we travel to work, shoe sizes and we finalize on creating a graph on our ages. Firstly, we created our own frequency table to check the numbers in each category by having us to place our individual magnets up on the board accordingly. In preschool setting however, its best to use their pictures of their faces to represent them on the graph. We then chart and plot our graph. There are various ways of plotting the graph. What my group did was to create a bar graph with a horizontal axis represting the number of people and the vertical axis representing the age group. Some of them drew out by shading the parts accordingly. Graphing a good topic we can start with in teaching younger children. From the given graph, they can compare which is more and which has less. 

I would like to thank Dr. Yeap for bringing all the information across clearly throughout this course. I definitely learnt a lot and have a better understanding to the different mathematical concepts.

Algebraic Thinking

Polygons, polygons, polygons
I learnt to find areas of the polygons through drawing out the geometry on a geoboard. Finding the area through a geoboard is rather interesting. The class has came out with different shapes and sizes on the geoboard. Usinga a small square as a standard unit of 1 square unit, we tried measuring the bigger polygons using the small square as a guide. Some polygons are easy to meaure while some polygons which has slanted lines are a little difficult to measure. Instead, we came out with ways such as cutting a smaller portion of it and placed it to the form a rectangle. Then it would be easier to measure the area of the bigger polygons. Maybe for younger children, I feel thatnwe should get them to ecplore by using rubber bands to form the different polygons on the geoboard first rather than to teach.the area. However, I find it rather confusing if a child were to change their focus into finding areas of poygons without using the geoboard.

Measurement & Geometry

Accommodation & Assimilation

Today I was taught about accommodation and assimilation. According to Piaget, accommodation refers to adaption process which involves altering one's schema for new information and experiences. Through assimilation, children take in new information and incorporate them to their existing ideas. It is important for teachers to know this, The implication here is that it is not good for adults to make every single thing crystal clear to young children for example spoon-feeding them with answers like what most parents do while guiding their children in their learning. I fell in love with the saying:-

"Everyone is made clever BUT not born clever"

Thirsty Crow
The class watched a short video called thirsty crow and it was indeed a very interesting story which teaches us about measurement of water. However, the difference of this problem that we have to solve involves fractions. We were supposed to find out how many liters of water left after Thirsty Crow drank 3 quarter of it. In all the methods discussed, I still find that it would be easier for me to use the methods where I made the denominator the same number hence, making it more easier to subtract. Although there are other methods, I find that it would be hard for young children to understand.

We were given time to experiment with the tangrams in making shapes. And surprisingly after cutting the shapes out, I was not able to put them back together to form the original shape before I cut them out. It was rather tricky to do so. Nevertheless, I learnt to use the various tangrams to visualize in getting the same area or even finding out what are the areas of the different shapes made using the tangrams given. 


Fraction is one of the mathematical concept that most young children faced in understanding. Even for myself, fraction is never an easy topic to learn since I was young. Epecially in solving word problem sums with fractions in it, I would simply just give up on it. It's a good thing that fraction is not taught in oreschool. I believe that young children need to know the basic mathematical concwpt first before moving on to fractions.

In class, we did an activity where we were told to fold a piece of paper into 4 equal parts and check if the fur parts are equal. How do we know that? We have to check if the 4 parts overlap with one another. Although we were told to fold the paper into 4 equal parts, we have actually came out with equal parts in many forms. Some of us folded the paper in a long rectangular where, while others folded in another rectagular way. Some even folded the paper in a triangular form. We looked through the various ways that we have come out with and we checked whether they are still equal parts although they are of different shapes by cutting and matching and overlapping them together.

I learnt the correct form of saying fractions for example 3 fourthd instead of 3 upon 4. Doing the word problem is not as easy as I think. I prefer solving the word problem through drawing out model so it would be easier for me to visualize clearly and solving through the problems.

Guessing, Estimation & Addition Story

Guessing & Estimation
I would love to carry out such guessing activity with children. Young children loves engaging activities like this. Getting them to make a guess will not only allow children to be engaged but also teaching them the mathematical concept in which I never thought it could be done. As I was doing the activity, I realized that I kept on shaking the can containing 3 paper clips and listened to the sound then comparing to the sounds frm the different cans. This is what we call a BENCHMARK to measure one thing against another. In order for young children to do such experiments, they should know the benchmark before making their guesses. For example, a child needs to know how heavy is 1kg if we were to ask qns like how heavy is your bag? They might think other measure as heavy as well.
When we get children to guess, it is best for teachers to take note of their language. Children need to know how things look like when we make use of comparative language like shorter, larger etc.

How to teach concept of big and small?
Concept is abstract through concrete experiences. When doing experiments or demonstration, teachers should ALWAYS start with things that are realistic instead of using manipulative. Children will understand better if they know in real life which is big and which is small. For example, an elephant is big and an ant is small. So it would be easier for them to understand the concept before we get them to compare. Then moving in to using manipulative.

Addition Story
There are different ways to create addition stories. Theynare subdivided into 3 parts together with the examples which we have discussed in class in cresting addition stories with (8+6).

1. Part-whole problem
E.g. Johnny has 8 marbles.
        James has 6 marbles.
        How many marbles do they have altigether?

2. Change Problem
E.g. Lisa has 8 pencils.
        Anne gives her 6 more pencils.
        How many pencils did Lisa have now?

3. Comparison Problem
E.g. Jess has 8 dollars.
        Bella has 6 dollars more than Jess.
         How much does Jess have?

Introduction to Mathematics & Use of Numbers

Ordinal Numbers         

 It was the first day of lesson. I was expecting a boring lesson, however, Dr. Yeap has proved me wrong. He made the introduction of the lesson very interesting. I didn't expect that names can be used as a Math activity. This indeed showed me a different perspective on looking at numbers. It was quite easy to be finding the 99th letter in the name. The first thing I noted when listing down the numbers was the pattern under the 3rd letter. However, it was not that easy as I thought when it comes to finding the 99th letter in my very own name. Ibwas not able to find the same pattern as I tried pn the previous name. Thus, I had to find other ways of solving the problem. And I managed to look at another pattern. From this activity, I learnt that there are many other ways to solve a problem. This applies to how we get young children to solve problem in class. There is no ONE way of solving a problem. Young children are great thinkers and they can actually come out with many different ways when we get them to solve a problem. I learnt that preschool teachers should never stop young children when you think they are doing the wrong method. Instead learn and facilitate them by modelling, scaffolding and providing examples. We all know that chidren would not understand explaining. Therefore, teachers should let and share with one another as Lev Vygotsku stated that children learn in a social situation. We use lots of visuals to teach young children like what Bruner mentioned to start lesrning in a concrete way. We preschool teachers should.also make use of CPA Approach (Concrete, Pictorial, Abstract Approach).

What role do numbers play?

Cardinal Numbers: Counting numbers to tell the smount & quantity
Ordinal Numbers : Counting numbers in terms of position and time
Measurement Numbers: Used to count uncountables lile money
Nominal Numbers: Literally saying out the numbers for e.g. we say 133 instead of one hundred and thirty-three

How to use numbers as cardinal numbers?
1 to 1 correspondence
Sorting & classifying
Rote counting: counting number in sequence
Rationale counting: counting for meaning, written as fraction or counted in ratio
Irrationale counting: examples of square root

Acceleration vs. Enrichment
Acceleration means to go to a next topic when children are able to do that particular topic.
Enrichment means to stay on the same topic BUT doing enrichment, giving a level of challenge on the task a little bit higher.

After this first session, I see math diffierently and hoped to know more on the other topics on mathematical concepts.

Friday, 29 March 2013

Exploring What It Means to Know and Do Mathematics

               How do we learn Mathematics? Jean Piaget and Lev Vygostky have answered them with their theories through the constructivism and sociocultural theory. Piaget mentioned about reflective thought where people connect existing ideas to new information where learners  construct new knowledge and tools to build on their prior knowledge. Vygostky stated about internalizing information within an individual's ZPD in which children are learning from their peers and adults. I like the saying about the construction of ideas in which not only children but even us as adults connects existing ideas to new ideas and giving a new meaning to it.

            During my time when I was learning Mathematics, memorizing formulas, time tables and so on is what I do best. But when it comes to understanding, I am totally weak in that. Why? The teachers did not make use of relational understanding where we were supposed to be given tools an manipulative or multiple representations to further understand the mathematical concept. Especially in young children, it is best to provide them with visual tools for them to explore and be involved. And although there are manipulative provided like what this chapter mentioned, they are ineffectively used. 

              I totally agree that using technology allows young generation to count. However, it may not be good to young children. Somehow I see young children nowadays do not use the concept I actually went through of counting mentally r even doing workings. Yet, it calculators help children work faster but it doesn't help children in generating their brain cells. I believe there are others ways for children to use in answering questions. It's because of the teachers who kept on wanting the right answers to the problem sums which led children so dependent on the use of technology.

              I totally agree that using technology allows young generation to count. However, it may not be good to young children. Somehow I see young children nowadays do not use the concept I actually went through of counting mentally r even doing workings. Yet, it calculators help children work faster but it doesn't help children in generating their brain cells. I believe there are others ways for children to use in answering questions. It's because of the teachers who kept on wanting the right answers to the problem sums which led children so dependent on the use of technology.

             An effective classroom for young children to be engaged in doing Math is essential. it should encourage children in risk-taking, reasoning, problem solving and so on. In my own teaching, I engage my children in productive struggle too as I believe that it is not wise of me to keep spoon-feeding them with answers. They will not be able to learn in that way and I ensure children that it is okay for them to make mistakes then from there they will learn. Children in my class have different abilities. Therefore, I created a classroom where their ideas and every discussions made are put up in class.

Teaching Mathematics in the 21st Century

             I remembered being taught the 5 content standards like geometry and measurement and even till now as a preschool educator, I'm teaching the students the same thing again. We can see that these content standards will remain even till the future. Not to forget, they are being taught in every level we are in too. Measurement is indeed a very popular topic and is greatly integrated in topics like science and language in young children's learning. 

            As the NCTM provides us with the Principles and Standards for School Mathematics, I believe that all teachers are making use of these six principles given  in educating young children in learning Mathematics. With these standards, every child is always given the opportunity to explore with the various mathematical concepts. Learning mathematics is not simple. I remembered being a child and my teacher just explaining the concept without any visual cues, I was definitely lost at that point of time. I feel that teachers should make use of the learning principle where teachers provide lots of experiences in the classroom making sure children understood the concept. Understanding the mathematical concept taught is very crucial as we know that math is everywhere. From my past experiences, it has lead me to have the positive attitude , persistence, and being reflective. This chapter explains it all and I believe strongly that by doing so, we will educate these young children better in not only understanding Mathematics but loving and enjoying the subject.

             However, I find that our Singapore government should work closely with the preschool teachers and understand mathematics for younger children. They have been stressing the children in this era in knowing the mathematical concept which are beyond their capabilities. It stated in this chapter that NCTM has  developed related standards documents about teaching where teachers made use of these resources in creating a classroom that supports teaching. However, when I flashback to how I learnt Math, I'm basically just trying to remember the formula and trying to get the grades to pass the exams without actually understanding them. Then what's the point of the major components that are needed to develop mathematical understanding when I don't see that happening in Primary and Secondary school as well.