Today marks the last day of our Elementary Mathematics module. I would like to say a big thank you to Dr. Yeap for making this journey of learning Mathematics a fun and fulfilling experience for me. To be honest, Math was never a subject I love but going through this module changed my perspective. I was surprised that I could actually do Math and yes, I am totally up for more Math experiences. I could tell that both Dr. Yeap and Peggy have put in a lot of effort in helping us learn Mathematics and I really appreciate that.

Today, we had a special guest speaker, Peggy Foo who came in to the class to share with us insights about learning Mathematics. Like what Dr.Yeap did (Bruner’s theory), Peggy used the Concrete materials to get us started then moved on to Pictorial and finally, the Abstract stage to solving a problem.

In the first problem, Peggy introduced us to the problem with an interesting opening. She was really good at it. Peggy pretended to be a magician and took out a set of play cards (large ones!) and got everyone involved by asking us to spell the number words. I thought that this lesson was somewhat unusual. It creates suspense and engaged the attention of the learners. Though the question (on how she could correctly arrange those cards in place) given was challenging, I was very motivated to find the answers. The challenges were just the right dose and we were eager to be the next ‘magicians’. The participation level was really high.

After experiencing the lesson myself, my top three take away from this lesson is that we should provide different approaches of learning (CPA) to help children of different ability / levels to learn. Secondly, we should give just the right dose of challenges for children. The problem should not be something that is too easy or difficult. The right dose of challenges will motivate and encourage them to want to work on the problem which leads to problem solving, cultivating thinkers. Lastly, teacher should try to make the lesson as interesting as possible. This would engage the learners and encourage them to learn.

This is another lesson we did today – visualization

My take away from this lesson:

1. Peggy introduced the ‘See’ , ‘Think’ and ‘Wonder’ table to the class. This thinking routine scaffold for one’s learning. It is a helpful tool as it breaks things down (step by step).

Wow! Today’s lesson was really challenging. I have no idea how to go about finding the angles. I just can’t seem to visualize. Hence, I was not able to recall which angle is equivalent to the other. Ha! No doubt, I was in such struggling stage. It was only later when my classmates pointed out the angles that were similar then I was able to proceed to the next level of solving the problem. With that help given, I tried to work out the problem in several ways but I was still not able to find the angle. Honestly, I was only able to ‘see’ the problem when I went through the steps that were given by my classmate. I went through each line of the equation and figure out the way why they did it that way. Having understand each line of equation, I finally got the understanding on how to derive to the answer.

Strangely, the algebra way of solving the problem helped me to visualize better. I was able to solve the question instantly by using the algebra method.

Okay, so I found out that if children (or even myself) find it hard to solve difficult problem like the one above, it is because they lacked the following:

1. Visualization (Develop the ability to visualize beyond what the eyes could see.)

2. Generalization

3.Metacognition

4.Number Sense

5. Communication (Able to present ideas on paper)

So how do you develop visualization ability?

Jerome Bruner says that:

‘the more young children move around, the better they are at visualization.’

‘the more concrete experiences you have in the early years, the more you would be able to visualize. Never compromise concrete materials in early Math.’

Although I cannot conclude how true the above statements are but I am sure that I was never an active child during my early years. Math was also not a subject that I could do well in. Jerome Bruner also talked about learning using the CPA approach. I certainly do not remember that my learning of Mathematics was through this approach. Learning Mathematics was always a stressful thing for me since young. From Dr.Yeap class, I got to understand how important AND IMPACTFUL early years Mathematics could have been. Having been through the ‘incorrect teaching or learning’ of Mathematics, I am determined to help children learn Mathematics the right way and that is none other than ‘Dr. Yeap approaches’.

Art and Math

Yes, now I see how I could use an art piece to come up with Mathematics lessons. It had never occur to me that I could use art pieces to write a Math problem. This is definitely something new and I am very interested to explore more about incorporating Art into Mathematics. Mathematics is all around us and yes, we could incorporate the learning of Mathematics into any subject or even through our daily routine care.

On a condition that : Any 2 tiles that touch must touch completely

So I worked on this problem with my peers. And wow! We came up with at least 9 figures…

BUT!

Dr. Yeap explained that if the figures can overlap, then they are the same.

Hence, that left us with only 2 figures…

In this video we count Polyominoes of different forms, and investigate some nice tiling problems. Some of these puzzles are very challenging, but fun even for primary school students.

Through this lesson, I came to realize that it is actually not difficult for educators to give children the concrete experiences to learn a Math concept. Any Mathematical concepts can be done through the CPA approach and it is really up to the individual educator to make the effort to make learning easy, fun and interesting for the children.

GEOMETRY

When Dr.Yeap posted the class the question:

Can you find the area of the polygon you have drawn?

My first reaction was ‘What? Find the area? How???

Then I began to think of ways in which I could tackle the problem.

1. I began to fall back on the typical area calculation using the formula method ‘Length x Breadth’ (No, it doesn’t work)

2. Oh there! I saw a square… oh no, wait, let’s see… if i try to turn the orientation of my triangle, oh yes… i can get another square. That makes two squares…and a triangle left. That’s 2 and a half square units.

Dr. Yeap then posted another question:

Is there any other method to find the area of the polygon you have drawn?

I studied the answers (areas of the 4 different polygons examples) on the whiteboard. I look at my paper, practically staring at the dots. Then I had an idea. Could it be the dots? I started counting. I have 5 dots. If I divide my 5 dots by 2, I would get an area of 2 and a half units. But is this the method to do it? (I went on to check if this ‘theory’ apply to the other examples.) To my surprise, IT DOES! I was in total awe.

It was an eye opener on how easily we could derive to the answer (the area of the different polygon) just by counting the number of dots. (Note: This theory only apply on the condition that there is one dot inside the polygon drawn.) It’s again about visualization isn’t it?

Then I was puzzled.

So, how can we find the area if I have more than one dot inside the polygon drawn?

Aha! That is when Dr.Yeap introduced Pick’s theorem to the class.

Pick’s theorem provides a relatively simple alternative. In order to use it, two definitions must be stated:

How many 1/8’s in 3/4? (Try solving this problem using the CPA approach)

Oh wait! I have a question. How do you know it’s 1/8?

– I can make eight equal pieces on the paper.

– Cut one of the pieces and start comparing. If it overlap, then the pieces are equal.

When children are ready, then move them on to the abstract form of learning:

‘It is important that we make sure children go through the journey (concrete and pictorial) to get to the destination. The journey that the children go through (process) is more important than the final destination (product).’ – Dr.Yeap

Today Dr. Yeap shared about linking Mathematics to other activities, apart from fiction books. Expose children to: Non-fiction books, Fine arts, Fiction books, Movies and Field Trips. It was really helpful when Dr. Yeap modelled to the class on how he could link movie to the in-class Mathematics lesson. In one of our lesson today, Dr. Yeap played a short clip of ‘Despicable Me 2’ and came out with a problem for us to solve. Personally, I thought that was a really clever idea. Not only did he manage to expose his students to the different activities to learn Math, it also catches our attention. Through this activity, I realized that when a teacher vary the sources used in class, it could make the lesson or activity come alive. I can’t wait to try this with the children!

I have minion/s with 2 eyes and minion/s with 1 eye.

In total, the minions I owned have 7 eyes.

How many minions do I owned?

(Hint: There can be more than one answer to this question. Good Luck!)

Today’s class is really interesting and have helped me in understanding the importance of being a knowledgeable and effective Mathematics educator. It was an eye opener to me that the ‘terms’ and the ‘questions’ that we have commonly used in class were inappropriate and it does not help children in learning Mathematics.

Example 1:

Can you solve this question?

+

Okay, so you must be thinking ‘What? Are you kidding me? Of course I DO KNOW the answer. I’m not a child, come on.’

But hang on! Did you think about this question –

Can we even add apples and oranges together?

When Dr. Yeap posted this question, I’m like ‘hey, wait! I have come across such questions way too often and why didn’t I even wonder or realize that these questions don’t even make sense in the first place. And how could it be possible that we could even come out with solutions? ‘

Learning Point: In the above example given, apple and orange are two different nouns. It is NOT possible to add two different nouns together.

Example 2:

Do we say:

(A) 7 is LESSER than 10

OR

(B) 7 is LESS than 10

Answer: B

Learning Point: LESS and LESSER is not the same word. You cannot use them interchangeably.

Lesser -> is the comparing of quality

Less -> is the comparing of quantity

Ten-Frames

The use of a ten-frame helps children build mental images and seeing quantities as instantly recognizable without having to recount from one. Hence, it gives children the CPA experiences! How wonderful can that be.

“You cannot imagine well what you haven’t experience” Dr. Yeap

I have to admit that I would usually use manipulative such as cubes to teach children counting. However, having being introduced to ten-frame, I would like to try it with the children in my class. Comparing the use of unfix cubes to ten-frame, I think that though cubes may be helpful in early stages of counting but it does not provide an instantly recognizable and distinct picture of the wholeness of ten like how ten-frame could achieve.

TIPS: The best part to ten frame is that you can either get them commercially or you can make them. Easy peasy!

Today marks the first class of Elementary Mathematics. I must say that today’s lesson was really an eye opener. Today, I came to experience myself the benefits of ‘productive struggle’. I am really glad that Dr. Yeap provided us time to work through (or struggle through) each activity within our group before providing the answer. The productive struggle indeed got us to be more motivated to tackle the problem. It was a really wonderful experiences for us to learn from one another. This learning is similar to Vygotsky’s theory of social constructivism where he states that children learn best in a group setting.

Dr. Yeap also brought in Bruner’s Theory of teaching using the following order:

1. Concrete

2. Pictorial

3. Abstract

Personally, I am a believer in teaching children with CPA approach. As an early childhood educator, I believe that children learn best through hands-on experiences hence, the need for use of concrete materials. In my classroom, children were given a variety of concrete materials (cubes, pencils, rubber bands etc) to explore with and to learn the different mathematics concept. Apart from using materials, I also get children to use their body parts to carry out their learning. It makes learning fun and interesting for the little ones especially when they are learning a new mathematical concept or idea.

I agree that a teacher can make or kill one’s interest for a particular subject. Through my own experiences, I agree that Math should be learn with understanding rather than ‘drilling’. In my secondary school days, my Math teacher would either drill his students with questions from ten years series or from stacks of worksheets from the various top schools. Over time, I felt like it was a chore to attempt the questions. I start to lose the eagerness to wanting to solve the math problem. The sense of satisfaction was never attained because learning becomes nothing more than ‘rote learning’.

In this reading, I would like to highlight two very interesting points which were discussed in the chapter.

Persistance, effort, and concentration are important in learning Mathematics.

This point helped me to be aware that ‘productive struggle’ is something which is beneficial to the students. In our local context, we seems to have many educators who feel the need to ‘explain’ or ‘show’ the answer to the students if we notice that they are spending more than the required time to complete the problem. To be honest, I am like the many educators out there who would go all in to explain to my student on a certain problem or task. However, through this reading, I realized that I am actually not helping my students instead, I am robbing away the much needed opportunities for them to understand Mathematics.

An unhurried setting provides children with the opportunities to self-explore and discover (Gonzalez-Mena, & Eyer, 2004).

Have students share and listen to each other.

In my experiences as an educator, I agree that having students share and listen to one another is a very effective strategy for learning, not particularly in just learning Mathematics but in other areas too. I observed that when children share their knowledge or findings with their peers, they are able to tap on these knowledge which invites further discussion into a topic. The sharing also helped children to identify, learn and look at things from a different perspective. Likewise in Mathematics, students can learn from one another that there are different strategies that they can use in approaching the same problem. This would help them to learn and understand that there is no one solution to solving a problem and as an educator, I welcome these variations.

Knowledge becomes more elaborate as children explore and interact with others (Geary & Bjorklund, 2000; Leslie, 2004).

In my own experiences with the nursery children in class, Math is a practical subject that we could use in our everyday life. For example, I had a child in the class who says ‘my mummy says I should share my biscuits with my friends. But mummy also says that I could eat 10.’ The child counts before she pops each biscuit into her mouth. That is Math and we are indeed using it so unknowingly each day. It goes to say that Math is a subject that could be used in any context and that learning Math could be a fun-filled experience if teachers adopt the right strategies and tools.