Lesson 6 : Card Trick

For today's lesson, I found out something interesting that I am very eager to learn. And that is the card trick, in which when I spell the word "O-N-E" sliding the card to the bottom of the pile (1 letter, 1 card) I am able to produce the card that has 1 in it. At first, when I saw the trick that Ms Peggy had done I was truly amazed. Of course, I didn't think it was magic but instead it was mathematics. I was eager to solve the card trick.

At first, I tried the odd, even, odd, even number pattern but could not solve the trick. But this does not stop me from trying to solve the trick. I tried to count O-N-E, and place the Ace card that represents one at the 4th position. At the 5th position, I counted T-W-O and placed the number 2 card at the 8th position. I understand that there are 10 cards in total, and the maximum position I can place the card is the 10th position. I carried on doing the remaining numbers till 10. And I placed the cards according to its position that I had placed, starting from the card at the 10th position at the bottom and pile the cards to the top, starting with the 1st position. I tried to do the card trick like what Ms Peggy did, and I was shocked myself that I had solved the card trick in which I can use this trick to amaze the 6 years olds' and get them to solve the card trick too !

Here's the line up of the cards starting from the 1st position on the left.

With this, children must have the prior knowledge of counting 1 to 10, knowing the values of 1 to 10, spelling the number words and recognition of numerical numbers 1 to 10.  Children are able to manipulate with concrete materials, in which in this case is the poker cards. 

This activity is challenging and interesting and I am sure the children would be eager to be engaged actively in this activity.

In addition, i have also learnt that when planning lessons for children, we must take note of the diverse learners in the classroom. As children learn at different pace, we could not "force" a child to learn when the child is not ready. If we "force" them to learn, they would give up easily as it is not within their capabilities. Instead, we should create activities for diverse learners, such that making an activity simple for struggling learners in which concrete materials must be used for them to enhance their visualizing skills. Whereas for the average learners, we should make the activity understandable for them. For the advance learners, if we make the lesson standardize for all they would get so bored and would end up getting into mischief. Instead, I should challenge them by getting them to do abstract activities in which involved them to think critically. However, the children should be learning the same content in which process and product may differ. This allowed us to be on the same track of content, of what we should be learning in that lesson. As children need prior knowledge of the content in order to do the task. 

I have also learnt to identify the "content", "process" and "product" of the planned lessons and this gives me a deeper insight of how i should plan my lessons in future to cater to diverse learners in my classroom. 

Lesson 5 : Trigonometry

Today, we touched on the topic of Trigonometry, as we study the relationship of a triangle's sides and angles between the sides. This topic was one of the topic I was weak at during my school days and upon confronting it today challenges me. As I realized that I am poor in visualizing a certain angle is the same as another angle of a the shapes of different length, and found it challenging to tackle the math problem. No matter how I look at the figure and tried hard to understand, I still could not get it. I realized the importance of visualizing and hope to not history repeats itself by enhancing the children under my care, abilities to visualize. 

We tried the math problem that is a primary 5 test paper, and was amazed of the complexity of the sums that would require much visualizing in order to solve the problem. With the topic of Trigonometry in mind, how does 5 years old children do, so by the time they reach 11, they are able to solve this math problem. One key word is to enhance the children's visualization ability.

i have also found out that with a triangle, by cutting out the angles, we can find out if it is 180 degrees by lining up the cut out angles in a row to see if it aligns.  This is helpful for me as it allowed me to have the hands on experiences to find out if the sum of the angles is 180 degrees in which I find it rather interesting. This is another aspect of visualizing.  

Through exploring objects and things around children, is one way to improve on visualization. As they observe the details of the object, such as the colors  lines, shapes. Mathematics learning enhance curiosity and enthusiasm of children as they grows naturally from their experiences of exploring with concrete materials. Preschoolers would learn to make mental images in their mind to refer to. While slightly older children learn to form shapes that they can move or change. However, children often have limited ideas about shapes as they lack the capability to think abstractly. Through this, the more children work on the geometric concepts, the more they can learn to explore spatial relationships. This would allow them to make and use simple maps as they describe the location in space.

Throughout this course, I have to learn the 5 ways to teach children mathematics through visualization, generalizing, number sense, metacognition and communication. Journal writing is one way to present the children's ideas of the concept as it is open ended and unlike worksheets, it is either right or wrong. 

Lesson 4 : Geometry

Today, I had learnt something new about the topic of Geometry. I can use geoboard to find the area of a figure, using the dots. This is something new to me, as I was only taught during my school days, that geoboards are only used for symmetry. Little did I know that actually the concept is about the same except that finding the area of the figure is much more abstract than symmetry in which I just need to find make a figure into equal parts or create another part to make a whole.

By introducing geometry to the children, get them to explore with the coloured square tiles by using a specific number of square tiles to form a figure, making sure that the sides touch one another and it must not be a repeated pattern.
This ensure that the children understand that with squares, we can produce other figures , grasping the concept of 1 square unit. 

By getting children to learn how to find the area of a figure, firstly, I need to show the children an example of 1 square unit. 

Then, I can get the children to explore with the concrete material, in which in this case is using a rubber band to create a figure that has a dot in it. When doing this activity, I wondered to myself "Why must we create a figure with a dot in it?" There must be something more to this. So I created a figure that has a dot in it.

By using 1 square unit, the children will be able to identify the area of the figure that they had created by counting the number of squares and visualize that one square is the same as 2 triangles. Thus, children must have the content knowledge of counting numbers, in which in this case is counting the numbers of square units, and understanding of shapes - 2 triangles = 1 square, 2 squares = 1 rectangle, and ability to visualize the different shapes that comprises of the figure.

As people think differently just like children do, there are different figures that were created that have a dot in it. With this, we found out that through drawing figures with one dot in it, area of the figures are different and thus perimeter are different too.

In class, we found out that in figures with one dot in it ...

5 dots  (perimeter) = 2 1/2 squares (area)

12 dots = 6 squares

9 dots = 4 1/2 squares

There is a number pattern in our findings in which ...

2 1/2 x _ = 5

2 1/2 = 2.5

5 / 2.5 = 2

Thus the number of dots is twice as much as the area in squares.

Out of curiosity, I actually tried to see if this formula applies the same to the rest of the number of dots in the figure.

I tried for 0 dot in the figure using a square with 4 dots ...

4 dots = 1 square

1 x 2 = 2

2 + 2 = 4

6 dots in a triangle = 2 squares

2 x 2 = 4 

4 + 2 = 6

I tried using 3 dots in the figure ...

15 dots = 8 squares

8 x 2 = 16

16 - 1 = 15

4 dots  in the figure

16 dots = 9 squares

9 x 2 = 18

18 - 2 = 16

5 dots in the figure,

18 dots = 10 1/2 squares

10 1/2 x 2 = 21

21 - 3 = 18

Through my findings, the formula does applies to the rest of the number of dots in the figures except that when there is 0 dot in the figure, I have to add 2 to the sum to match the total number of dots.

However, as I increased 1 dot in the figure, the number gets bigger but subtracting it instead.

3 dots = -1

4 dots = -2

5 dots = -3

I find this activity rather interesting, as it challenges my ability to think in depth as I wondered why must we put 1 dot in the figure, and does this apply the same to the number of dots as I increase or decrease the number of dots in the figure. 

With this, I managed to solve this problem with abstract thinking and children were able to do this, provided they are given the opportunity to explore with the concrete materials and picture it through visualization.

Lesson 3 : Fractions

Through today's session, I had made "friends" with fractions. As I finally understood the reason behind the equation of 3/4 divide 1/2.

Fractions is meant by a part of a whole, but children would not be able to understand this unless they visualize how fractions work.

As children learn through exploration of concrete materials, by getting them to fold the paper into four equal parts and shading 3/4 of the paper is the first step of getting them to do hands on activity as an introduction to the lesson of fractions. 

The next step is to use diagrams to allow children to use pictorial aids to help them in their understanding of fractions. This is something new to me, as I always thought bar graph method is only used for problem sums. Never did I realized that bars can help me understand fractions too !  As I often associate pizza with fractions and I'm pretty much "afraid" of fractions as to me, it is rather complicated ! However, through today's lesson prove me wrong ..

How many are shaded?

With a glance, i do not need to count how many is being shaded,

and the answer is 2 1/4. This is because I see 2 whole parts are shaded thus giving me the answer of 2 whole number, while 1 part out of the 4 parts are shaded in which is the same length as the other 2 bars. Thus I concluded that 

2 1/4 are shaded.

Through this diagram, by thinking of different ways to take away 

1/2 from 2 1/4, it helped children to think abstractly.

As they had already explored with the concrete material and in this case, is the folded paper and shaded parts in it. And moved on to the pictorial aids by drawing the diagram. Now, by getting children to think abstractly, they are able to work out the math solutions.

In which for my case, I used the method of 

2 1/4 - 1/2 

= 9/4 - (1/2 x 2)

= 9/4 - 2/4

= 7/4

As 4/4 makes a whole, 7 - 4 = 3, thus it's 1 3/4.

Why do i need to complicate things further? It is actually simple as it looks !

The answer is actually in the diagram itself, I just need to draw some lines to indicate the half is being taken away.

Strike off 1/2 of 1 whole, and dissect it into 4 equal parts.

Rearrange the parts by placing the 1 part with the 1/2 to give a clearer picture.

And it is clearly stated 1 3/4 through the yellow shaded bars.

When I found out this new concept, I was relief of the stress as never did I thought it was actually this easy ! Through this, children are able to understand the concept of taking away 1/2 from 2 3/4.  Bid goodbye to complicated fractions !

Now I feel I understand fractions and I'm gonna make fractions my best friend and not to be afraid of it no more! 

I have googled on the web for DIY teaching aids to help children understand fractions better..

This can be made by the children as they engaged in concrete materials to understand fractions. 1 whole can be spilt into different parts.

Lesson 1 : Tangram

When I was given the tangram pieces, and was asked to solve the problem to make a rectangle using the tangram pieces, I tried to make a rectangle using as many tangram pieces as possible. I explored with the tangram pieces, by fixing the pieces together using the trial and error method. There are a mixture of shapes - square, triangles of 3 sizes (big,medium,small) and parallelogram.  

Dr Yeap got us to make a rectangle using the tangram pieces, and I managed to create a rectangle rather quickly as there are no specific number of pieces i should use.

With the specific number of pieces I should use in order to create a rectangle, require much logical and critical thinking skills. As i need to try and foresee which tangram pieces I can used in order to fit into a rectangle.

I managed to create a rectangle using ...

3 pieces tangram

4 pieces tangram

This is a 5 tangram pieces to make a rectangle. This was the easiest to do as i just made a square with 2 triangles and created a mini rectangle using 2 small triangle and 1 parallelogram.

When figuring what other ways to make a rectangle. I found out that there are actually other ways to make a rectangle using, 5 tangram pieces. And I have found 2 more other ways to create a rectangle using the tangram pieces.

 

This is the 6 tangram pieces 

And the 7 tangram pieces in which all pieces are used !!

The children need to know what is a rectangle, and how does it looks like, to do this activity. Through this activity, the children will be able to understand that shapes can be rotated, and in regardless of he orientation, a rectangle is still a rectangle. Children were also given the opportunities to explore with shapes and needed to be challenged with complex solutions - can you make a rectangle using all of the tangram pieces? This will definitely push the children to a whole new level ! When I ask myself this question, I thought it was impossible. But nothing is impossible, is a matter of whether would you want to do it. As I started doing the 3 pieces tangram, and advancing to the next piece and finding other ways to make a particular number of tangram pieces to form a rectangle, it gives me a sense of satisfaction and achievement. And it almost felt addictive to just solve the solution. 

Lesson 2 : Whole Numbers

Through today's lesson, I learnt of a new math concrete material, 10 Frames. Never did I realized that such a simple material can be made to be a math manipulative for the children to explore with whole numbers.

What exactly is 10 frame? It is a frame with 10 boxes !

I googled on do-it-yourself 10 frame, and had found an alternative way of making my very own 10 frame. And that is by using egg cartons, and label the numbers from 1 - 10 on each egg holder.

With the theme of "Jack and the beanstalk", I was given beans to manipulative with my DIY 10 frame. The beans was used as an actual object that is related to the theme of the lesson. This allowed children to use actual things to count with. With Dr Yeap's open ended questions, it allowed me to think of the way he used words wisely to specifically ask the questions in detail.

"5 Beans is half of 10. How do you know?"

I know 5 beans is half of 10 because, 5 boxes are empty while the other 5 boxes are filled with beans. This made an equal amount of number. 

Through this activity, it enhances my critical thinking, and it allowed me to think in depth. With Dr Yeap's questions, "I wonder why ... ", "How do you know ... " it allows me to bring my curiosity to a whole new level. It made me wonder the reason behind certain things and challenge myself to do more than i could ever think of.

With the concrete materials, it helped me to visualize the problems better and work the solution with ease. I believe this will also work well with children too, so as to enhance their visual skills by seeing numbers visually.

Upon introducing this activity to the children, i have to ensure that the children have the basic knowledge of knowing and understanding the value of 1 - 10. With 10 frame, it also helped children to learn the different combination of number bonds, up to 10 ( 7 is more than 3, 4 + 2 make up a 6, 6 is double of 3, 6 is 4 less than 10). 

No matter how each individual arranges the beans on the 10 frame, 7 is still 7. Children are also able to enhance their 1 to 1 correspondence skills when they place the beans into each boxes. They are able to learn number facts through enhancing their counting and addition, subtraction skills. 

This allowed the children to develop subitization, in which they are able to identify the number of objects in a set without counting. With practices, the children will be able to recognize the number at a glance of the arrangement. In addition, children will be able to identify the number of empty boxes which is the number complement to 10.

I can also extend the activity by adding another 10 frame when the children are familiar with the basic of counting the beans using one 10 frame. This allowed the children to enhance their number sense, by exposing to numbers from 1 - 20. Through this, they are also able to enhance their knowledge of +9 in which is the same as adding 10 and removing 1. 

With this, I would like to end this blog post with a video that shows an independent learning activity of using 10 frame that I would love to add into my classroom!

Pre-course Reading on Chapter 1 & 2

Time passed.

Society had changed ... 

In order for us, teachers to move forward in trend with the students, our teaching are consistently improving. This is for the benefits of our future younger generations, especially in our case as we are the first to touch the young hearts and to inspire them.

As an Early Childhood Educator, I strongly believe in "leaving no child behind". Every child should have strong support and opportunities to learn math as math is essential in our daily life, unknowingly. Imagine how hard life would be, when you step into the lift and could not identify which number to press, for you to get home. 

Math is no longer just about 123... Instead, it is a perfect combination of Math and Science. Science is defined as a system of acquiring knowledge, whereby observations and experimentation are used to explain the reasons. While Math is defined as concepts and methods that have a regular pattern and sequence. 

By getting children to observe the concept and experiment the trial and error method in order to solve the solution is okay. Never underestimate a child's potential. There are like 835536 ways to solve a problem, and the essential point is to find the suitable concept to solve it. 

We should move away from spoon feeding the children with concepts, and expecting them to throw it out back during tests. Give the children ample space and time for them to work out the solutions on their own. Let them understand the concept, and solve the solution with their own methods. Allow them to explore Math. It doesn't hurt to be innovative ain't it? 

Say goodbye to drilling methods and teacher-directed lessons.

Who says Math can't be fun ?