31st August 2011

3 Interesting Things

1) 4 divided by 2/3

This is the toughest problem I've encountered throughout this course. My brain just doesn't get what does this equation means, and the only method I can use is to make 4/1 times 3/2 (what I was taught in primary school). However, upon using the coloured paper and seeing the fractions in pictorial form on the coloured papers, I can understand what this equation means. Sometimes, the simplest things are made complex by the human mind.

2) MRT Station

This was an exciting excursion for us. It showed us that maths is all around us!

My group's answer:

We measured one step to be 15cm.

There are 4 sets of stairs in total, and each set has 16 steps.

16 X 4 = 64

There are 64 steps.

64 X 15 = 960cm.

The total height is 960cm.

3) The most appropriate-sized container to fit 15 beans

From this activity, I realized we really underestimated the volume of a container and how much it can contain. Perhaps the number 15 was comprehended to be quite a big number, which explains why many of us made a container that size when in actual fact it is much to big. And I still thought that my group's container was really small and I was even wondering if it could fit 15!

Reflection:

Maths is not always what it seemed to be - boring and lifeless. Maths is all around us, for example the MRT station activity, even just shapes around us and it is a subject that is logical and simple in its complexity. I have learnt that maths can be made fun through the activities we teachers conduct in class, and it all comes down to how we teach a concept and not just drilling or formulas to be spoon-fed. This is my first experience of enjoying maths at all, in all my years of schooling. Thank you Dr Yeap, for planning a continuous array of activities that really enhanced our mathematics experiences! I like the way you immediately think of an interesting activity to do even though it was a spontaneous topic or thought someone brought up! You know your stuff really well and convey them interestingly! :)

# EDU 330 Elementary Mathematics

## Friday, September 2, 2011

### 6th Lesson: 3 interesting things

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Never believe that a few caring people can't change the world. For, indeed, that's all who ever have. - Margaret Mead
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## Thursday, September 1, 2011

### 5th Lesson: 3 interesting things

26th August 2011

3 Interesting Things

1) The 'Dot' Pattern

This is the first time I have heard of the Pick's Theorem. I have to admit it was introduced in a really interesting way though - by getting us to draw as many different sized squares as possible. And I was pretty surprised by myself as I could figure out more than 6 ways to draw the different squares!! Calculating the area was tough, I was extremely impressed with the "add the number of dots minus three" relationship that my classmate came up with! Kudos to her!!

2) The Graph Chart

Graph is my favourite topic especially when teaching children. It sparks interest in them and they can all participate and see their "cube" on the graph chart. The graph can be anything from "My Favourite ______" to "How I Travel To School" etc. However, it was a challenge to come up with a graph without proper instructions (look at how used we are to listening to instructions), but we quickly decided to come up with what we could!

3) Quiz Question

"Monsters can have 3 eyes or 2 eyes. How many monsters are there if they have 19 eyes altogether?"

My first instinct was to do trial and error method, as I was taught in school. However my intelligent partner pointed out later on, that Dr Yeap had taught us number bonds! From 19, 10 and 9 can be taken out and 10 can be divided by 2 while 9 can be divided by 3. Altogether it would be 5 + 3 which would make the answer 8 monsters altogether!

Reflection:

Overall, it was a really interesting lesson although the topics were familiar. Yet they felt unfamiliar. Probably this is evidence that we are exploring the same topic from a different angle which is really crucial in any good curriculum. To make the familiar unfamiliar. (:

3 Interesting Things

1) The 'Dot' Pattern

This is the first time I have heard of the Pick's Theorem. I have to admit it was introduced in a really interesting way though - by getting us to draw as many different sized squares as possible. And I was pretty surprised by myself as I could figure out more than 6 ways to draw the different squares!! Calculating the area was tough, I was extremely impressed with the "add the number of dots minus three" relationship that my classmate came up with! Kudos to her!!

2) The Graph Chart

Graph is my favourite topic especially when teaching children. It sparks interest in them and they can all participate and see their "cube" on the graph chart. The graph can be anything from "My Favourite ______" to "How I Travel To School" etc. However, it was a challenge to come up with a graph without proper instructions (look at how used we are to listening to instructions), but we quickly decided to come up with what we could!

3) Quiz Question

"Monsters can have 3 eyes or 2 eyes. How many monsters are there if they have 19 eyes altogether?"

My first instinct was to do trial and error method, as I was taught in school. However my intelligent partner pointed out later on, that Dr Yeap had taught us number bonds! From 19, 10 and 9 can be taken out and 10 can be divided by 2 while 9 can be divided by 3. Altogether it would be 5 + 3 which would make the answer 8 monsters altogether!

Reflection:

Overall, it was a really interesting lesson although the topics were familiar. Yet they felt unfamiliar. Probably this is evidence that we are exploring the same topic from a different angle which is really crucial in any good curriculum. To make the familiar unfamiliar. (:

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Never believe that a few caring people can't change the world. For, indeed, that's all who ever have. - Margaret Mead
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## Sunday, August 28, 2011

### 4th Lesson: 3 interesting things

25th August 2011

3 Interesting Things

1) Mind-Reader Game

3 - 27

4 - 36

5 - 45

6 - 54

7 - 63

8 - 72

Variety of methods for the “mind reader” to work out the person’s number.

Multiply by 10 then minus the 1st number that the person says. For example: 3 would be 30 – 3 = 27.

If the person say 8,

Steps:

8 and 2 put it together as 82

82 - 8 - 2 =72

Multiples of 9.

2) Variation of Word Problems

There are 37 children. 19 are boys. The rest are girls. How many girls are there?

-> This is a part-whole situation because there are 2 parts and a whole.

There are 37 monsters in a game. I shot down 19 monsters. How many are left?

-> This is a change situation because there is a before and after quantity and a final quantity.

The highest level would be the comparison situation because there are 2 quantities being compared.

3) Fractions taught through coloured paper

What struck me during this lesson was that fractions can be taught through the use of such coloured paper and many 'complex' concepts to children could be made simple through the smart usage of these. This is definitely something I would try out with my children!

Reflection:

Being exposed to the various word problems and knowing the variations that should be present for a child to fully grasp the concept of whether it is addition or subtraction etc, it would aid the teacher in understanding a child's mathematical development because of this knowledge. I love the first 'mind reader' game and have tried it out with my children. It has motivated them to learn maths in an interesting way because they feel maths is magic! They strive to figure out the answers and see the pattern in it. It keeps their brain gears going and keeps them brainstorming on different ways to solve these problems.

3 Interesting Things

1) Mind-Reader Game

3 - 27

4 - 36

5 - 45

6 - 54

7 - 63

8 - 72

Variety of methods for the “mind reader” to work out the person’s number.

__1st Method__Multiply by 10 then minus the 1st number that the person says. For example: 3 would be 30 – 3 = 27.

__2nd Method__If the person say 8,

Steps:

8 and 2 put it together as 82

82 - 8 - 2 =72

__3rd Method__Multiples of 9.

2) Variation of Word Problems

There are 37 children. 19 are boys. The rest are girls. How many girls are there?

-> This is a part-whole situation because there are 2 parts and a whole.

There are 37 monsters in a game. I shot down 19 monsters. How many are left?

-> This is a change situation because there is a before and after quantity and a final quantity.

The highest level would be the comparison situation because there are 2 quantities being compared.

3) Fractions taught through coloured paper

What struck me during this lesson was that fractions can be taught through the use of such coloured paper and many 'complex' concepts to children could be made simple through the smart usage of these. This is definitely something I would try out with my children!

Reflection:

Being exposed to the various word problems and knowing the variations that should be present for a child to fully grasp the concept of whether it is addition or subtraction etc, it would aid the teacher in understanding a child's mathematical development because of this knowledge. I love the first 'mind reader' game and have tried it out with my children. It has motivated them to learn maths in an interesting way because they feel maths is magic! They strive to figure out the answers and see the pattern in it. It keeps their brain gears going and keeps them brainstorming on different ways to solve these problems.

Posted by
Never believe that a few caring people can't change the world. For, indeed, that's all who ever have. - Margaret Mead
at
12:34 PM
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## Thursday, August 25, 2011

### 3rd Lesson: Lesson Study

24th August 2011

It was an engaging discussion learning about "lesson study".

I like how the lesson was paced, by getting us to talk about what we know about lesson study first, then exposing us to the gist of lesson study by showing us 2 case studies and then how it can benefit us as teachers professionally.

The 2 case studies were interesting, I especially enjoyed watching our guest lecturer take a class of K1 on the cubes activity (: One thing I took away from that video was her enthusiasm and her questioning ability! Personally I feel a teacher can never have too many questions as questions stimulate thinking and may not necessarily require an answer, rather, just to get the brain gears going.

I learnt that besides our norm here, which is attending workshops or conferences, milestone courses, having informal/formal mentoring during in-house training and reading up professional development books and articles, alternatives like lesson studies may also work well. In our context, some may not be as open nor comfortable with having other professionals sit in our classrooms and observe us, but it is undeniable that good points and learning do take place because it is important that we improve and perfect our practices from time to time. This is shown and proven through the videos shown, in reference especially to the one in Japan context.

All in all I have learnt something new and it was through an engaging method, and no I did not fall asleep! Hooray!

It was an engaging discussion learning about "lesson study".

I like how the lesson was paced, by getting us to talk about what we know about lesson study first, then exposing us to the gist of lesson study by showing us 2 case studies and then how it can benefit us as teachers professionally.

The 2 case studies were interesting, I especially enjoyed watching our guest lecturer take a class of K1 on the cubes activity (: One thing I took away from that video was her enthusiasm and her questioning ability! Personally I feel a teacher can never have too many questions as questions stimulate thinking and may not necessarily require an answer, rather, just to get the brain gears going.

I learnt that besides our norm here, which is attending workshops or conferences, milestone courses, having informal/formal mentoring during in-house training and reading up professional development books and articles, alternatives like lesson studies may also work well. In our context, some may not be as open nor comfortable with having other professionals sit in our classrooms and observe us, but it is undeniable that good points and learning do take place because it is important that we improve and perfect our practices from time to time. This is shown and proven through the videos shown, in reference especially to the one in Japan context.

All in all I have learnt something new and it was through an engaging method, and no I did not fall asleep! Hooray!

Posted by
Never believe that a few caring people can't change the world. For, indeed, that's all who ever have. - Margaret Mead
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## Wednesday, August 24, 2011

### 2nd Lesson: 3 interesting things I learnt and 1 funny fact

23rd August 2011

3 Interesting Things

1) The 'Dice' Game

Put two dice together, just by looking at the numbers at the two opposite ends can you guess what is the total of the two numbers in between the two sides put together?

Thoughts: This game really needed some 'number sense' to solve it. The trick is that the total of the 4 numbers from the two sides put together and the 2 opposite ends would always add up to 14.

2) A 2-digit number - A 1 digit number = 3 (or any other number)

(above picture taken from Dr Yeap's blog)

Thoughts: It seems that it was a stroke of brilliance that the last number shows the number of equations that could be made with that number as an answer. I totally could not see the pattern until someone pointed it out to me (Demonstrating a lack of number sense once again).

3) What is Mathematics?

-Generalization

-Communication

-Metacognition

-Number sense

-Visualization

Funny Fact

In regards to the model method taught in primary school, I like how Dr Yeap put it - "It is taught as a vehicle to manage information" because none of us really use these models to solve real life daily problems on how much money John has got altogether etc. We will instead, ask John how much he has got and if he allows, take his wallet and count it for him.

Hahahaha!

Reflection:

I like the fact that math is being taught in a way that stimulates thinking now, instead of the olden days whereby memory work makes up a large component. In Dr Yeap's words, "The times table has violated the learning principle as it is inappropriate." Instead, the MOE has adopted the Concrete -> Pictorial -> Abstract (CPA approach by Jerome Bruner) which teaches children how to figure things out instead of focusing on memory work. Also, I like how Dr Yeap recognizes that "the ability to remember is not a human strength" because this is something many adults around me has denied while I was growing up and I was often severely punished for not being able to remember how much does 13 X 13 adds up too.

3 Interesting Things

1) The 'Dice' Game

Put two dice together, just by looking at the numbers at the two opposite ends can you guess what is the total of the two numbers in between the two sides put together?

Thoughts: This game really needed some 'number sense' to solve it. The trick is that the total of the 4 numbers from the two sides put together and the 2 opposite ends would always add up to 14.

2) A 2-digit number - A 1 digit number = 3 (or any other number)

(above picture taken from Dr Yeap's blog)

Thoughts: It seems that it was a stroke of brilliance that the last number shows the number of equations that could be made with that number as an answer. I totally could not see the pattern until someone pointed it out to me (Demonstrating a lack of number sense once again).

3) What is Mathematics?

-Generalization

-Communication

-Metacognition

-Number sense

-Visualization

Funny Fact

__The 'Model' Method__In regards to the model method taught in primary school, I like how Dr Yeap put it - "It is taught as a vehicle to manage information" because none of us really use these models to solve real life daily problems on how much money John has got altogether etc. We will instead, ask John how much he has got and if he allows, take his wallet and count it for him.

Hahahaha!

Reflection:

I like the fact that math is being taught in a way that stimulates thinking now, instead of the olden days whereby memory work makes up a large component. In Dr Yeap's words, "The times table has violated the learning principle as it is inappropriate." Instead, the MOE has adopted the Concrete -> Pictorial -> Abstract (CPA approach by Jerome Bruner) which teaches children how to figure things out instead of focusing on memory work. Also, I like how Dr Yeap recognizes that "the ability to remember is not a human strength" because this is something many adults around me has denied while I was growing up and I was often severely punished for not being able to remember how much does 13 X 13 adds up too.

Posted by
Never believe that a few caring people can't change the world. For, indeed, that's all who ever have. - Margaret Mead
at
11:23 AM
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## Tuesday, August 23, 2011

### 1st Lesson: 3 interesting things I learnt and 1 funny fact

22nd August 2011

3 Interesting Things

1) Learnt the "ten-frame" which I really found that it would be really useful for children - clear-cut and concrete.

2) Learnt the 'spelling' trick with cards. Will perform this to my children in hope to spark an interest in them towards numbers!

3) Arrange five numbers. This looks simple but I found it really complex, whether in regard to my past failures in mathematics or it is really complicated. But I understood in the end, amazingly (: Sense of accomplishment!

Funny Fact

Teachers showing a picture of three (or more) boys before the finishing line and asking

"Who is the third in the race?" (testing on Ordinal Numbers)

Answer: This question can't be answered (phrased incorrectly!) as they have not completed the race yet! Hahaha..

Reflection:

Surprisingly, the first session wasn't as boring as I expected. I always enjoyed having a teacher with a good sense of humour especially in a subject such as mathematics (: Interesting problems were posed and I admit, I struggled with all the problems but it is the struggle that adds/turns into the sense of accomplishment when the problem is finally solved. I was really influenced by all the positive energy moving around in the class, and inspired to look for other strategies and ways after meeting a dead-end. All in all, a fulfilling session that really used up my brain cells, resulting in me sleeping really soundly yesterday night. Hahaha!

3 Interesting Things

1) Learnt the "ten-frame" which I really found that it would be really useful for children - clear-cut and concrete.

2) Learnt the 'spelling' trick with cards. Will perform this to my children in hope to spark an interest in them towards numbers!

3) Arrange five numbers. This looks simple but I found it really complex, whether in regard to my past failures in mathematics or it is really complicated. But I understood in the end, amazingly (: Sense of accomplishment!

Funny Fact

Teachers showing a picture of three (or more) boys before the finishing line and asking

"Who is the third in the race?" (testing on Ordinal Numbers)

Answer: This question can't be answered (phrased incorrectly!) as they have not completed the race yet! Hahaha..

Reflection:

Surprisingly, the first session wasn't as boring as I expected. I always enjoyed having a teacher with a good sense of humour especially in a subject such as mathematics (: Interesting problems were posed and I admit, I struggled with all the problems but it is the struggle that adds/turns into the sense of accomplishment when the problem is finally solved. I was really influenced by all the positive energy moving around in the class, and inspired to look for other strategies and ways after meeting a dead-end. All in all, a fulfilling session that really used up my brain cells, resulting in me sleeping really soundly yesterday night. Hahaha!

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at
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## Saturday, August 20, 2011

### Chapter 2

Upon reading Chapter 2, I have identified two concepts among all that I recognize of utmost importance. Hence, I shall summarize and reflect on what I have learnt upon understanding them.

The first concept would be

The Setting for Doing Mathematics:

To regard all ideas expressed by anyone in equal respect as it has the potential to contribute to everyone's learning.

Students to understand that methods have to be understood by others and also, there is no 'one-method' in solving a problem.

Mistakes can be dealt with constructively, instead of hiding it. They are opportunities for furthur exploration in reasoning and problem solving.

Embrace the culture of learning - Allow students to come up with individual approaches rather than being dependent on the teacher.

Opinion:

I selected this concept of "The Setting for Doing Mathematics" as I felt that this is something I can identify with, especially with the particular point that the classroom culture exhibits an appreciation for mistakes as opportunities to learn.

My personal experience when I was younger was that if I made mistakes, my mathematics workbook would be thrown to the ground and I would have to pick it up and redo the problems on my own until I get the correct answer. On the contrary, students who get the problems right could have their free time to chit chat, do drawing or their own activities. Hence, I quickly learnt that the product rather than the process have to be right. I often 'covered up' my own mistakes by erasing my answer and writing in the correct ones (that majority of my friends had) before handing it up. I recognize that this is not a strategy to be used in class because it teaches students the wrong concepts. This is an example of how a student has to feel safe to make mistakes in the classroom environment in order to be a learner of depth.

********

The second concept would be

Implications for Teaching Mathematics:

Students should have the leeway to develop, or invent, strategies for doing mathematics using their prior knowledge - demonstrating comprehension between the connections of these maths concepts.

To create an environment in which students interact with each other and the teacher, leading to reflective thinking.

Students to find relevant ideas from a new idea in order to bear the development of the new idea.

Provide opportunities for students to build connections between what they know and what they are learning.

Students may make errors as they misapply their prior knowledge to what they have learnt. These errors are opportunities for learning as teachers can scaffold and lead students to engage in self-reflective thought process.

Scaffolding in mathematics requires use of tools like manipulatives or more assistance from peers.

Each learner is unique and possess a different collection of prior knowledge and cultural experiences. Effective teaching incorporates what each student bring to the classroom.

Opinion:

Similar aspects I identified that re-appeared in implications for teaching mathematics are "Treat errors as opportunities for learning" and "honor diversity", both points which were presented differently in 'The Setting for Doing Mathematics' while in essence, the same.

I like how the author puts it, that errors are misapplication of student's prior knowledge in a new situation. This makes sense in a way that learners are continuously tapping onto their prior knowledge when solving problems, and when faced with a challenge their errors are most probably because they do not understand what is the most useful approach to use for the new situation. These are misconceptions that the teacher use it as opportunities for the student to engage in reflective thought.

Honoring diversity was also highlighted in 'The Setting for Doing Mathematics' as "Ideas are the currency of the classroom". Technically, the essence of both factors is for the teacher to respect and honor the ideas/prior knowledge that unique individuals bring into the classroom. Effective teaching can then be done as new ideas taught are built onto those prior experiences. Certainly all the factors are important as they all play a role, but I feel that factors that re-appear are of utmost importance in the learning and teaching experience for both learners and teachers.

Reference:

Van De Walle, J., Karp, K. & Bay-Williams, J. (2010). Elementary & middle school mathematics. Teaching developmentally (7th ed.). Boston, MA: Allyn and Bacon

The first concept would be

The Setting for Doing Mathematics:

__Ideas are the currency of the classroom__To regard all ideas expressed by anyone in equal respect as it has the potential to contribute to everyone's learning.

__Students have autonomy with respect to the methods used to solve problems__Students to understand that methods have to be understood by others and also, there is no 'one-method' in solving a problem.

__The classroom culture exhibits an appreciation for mistakes as opportunities to learn__Mistakes can be dealt with constructively, instead of hiding it. They are opportunities for furthur exploration in reasoning and problem solving.

__The authority for reasonability and correctness lies in the logic and structure of the subject, rather than in the social status of the participants__Embrace the culture of learning - Allow students to come up with individual approaches rather than being dependent on the teacher.

Opinion:

I selected this concept of "The Setting for Doing Mathematics" as I felt that this is something I can identify with, especially with the particular point that the classroom culture exhibits an appreciation for mistakes as opportunities to learn.

My personal experience when I was younger was that if I made mistakes, my mathematics workbook would be thrown to the ground and I would have to pick it up and redo the problems on my own until I get the correct answer. On the contrary, students who get the problems right could have their free time to chit chat, do drawing or their own activities. Hence, I quickly learnt that the product rather than the process have to be right. I often 'covered up' my own mistakes by erasing my answer and writing in the correct ones (that majority of my friends had) before handing it up. I recognize that this is not a strategy to be used in class because it teaches students the wrong concepts. This is an example of how a student has to feel safe to make mistakes in the classroom environment in order to be a learner of depth.

********

The second concept would be

Implications for Teaching Mathematics:

__Build new knowledge from prior knowledge__Students should have the leeway to develop, or invent, strategies for doing mathematics using their prior knowledge - demonstrating comprehension between the connections of these maths concepts.

__Provide opportunities to talk about mathematics__To create an environment in which students interact with each other and the teacher, leading to reflective thinking.

__Build in opportunities for reflective thought__Students to find relevant ideas from a new idea in order to bear the development of the new idea.

__Encourage multiple approaches__Provide opportunities for students to build connections between what they know and what they are learning.

__Treat errors as opportunities for learning__Students may make errors as they misapply their prior knowledge to what they have learnt. These errors are opportunities for learning as teachers can scaffold and lead students to engage in self-reflective thought process.

__Scaffold new content__Scaffolding in mathematics requires use of tools like manipulatives or more assistance from peers.

__Honor diversity__Each learner is unique and possess a different collection of prior knowledge and cultural experiences. Effective teaching incorporates what each student bring to the classroom.

Opinion:

Similar aspects I identified that re-appeared in implications for teaching mathematics are "Treat errors as opportunities for learning" and "honor diversity", both points which were presented differently in 'The Setting for Doing Mathematics' while in essence, the same.

I like how the author puts it, that errors are misapplication of student's prior knowledge in a new situation. This makes sense in a way that learners are continuously tapping onto their prior knowledge when solving problems, and when faced with a challenge their errors are most probably because they do not understand what is the most useful approach to use for the new situation. These are misconceptions that the teacher use it as opportunities for the student to engage in reflective thought.

Honoring diversity was also highlighted in 'The Setting for Doing Mathematics' as "Ideas are the currency of the classroom". Technically, the essence of both factors is for the teacher to respect and honor the ideas/prior knowledge that unique individuals bring into the classroom. Effective teaching can then be done as new ideas taught are built onto those prior experiences. Certainly all the factors are important as they all play a role, but I feel that factors that re-appear are of utmost importance in the learning and teaching experience for both learners and teachers.

Reference:

Van De Walle, J., Karp, K. & Bay-Williams, J. (2010). Elementary & middle school mathematics. Teaching developmentally (7th ed.). Boston, MA: Allyn and Bacon

Posted by
Never believe that a few caring people can't change the world. For, indeed, that's all who ever have. - Margaret Mead
at
12:43 AM
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