Lecture #5.

The Transformer in math - Reflect, stretch, shear, translate, rotate... it almost sounds like aerobics for shapes, too.

Tonight's class, we were asked to draw as many squares as we can within the dots. Coming up with 6 was not difficult... The next task was to draw any shape but only using 4 dots to connect. And coming up with that was exciting because we had to visualize in different ways from the previous activity.

We made another discovery that night that has actually already been discovered way before us. And that theorem was made by George Pick.

To cut a long story short, George Pick discovered a relationship between the area of a simple
polygon, the number of dots on the boundary of the polygon and the count points inside the polygon. We were trying to figure what exactly that relationship was in class tonight. While George Pick came up with a theory and I would assume a formula for this relationship as well, I think I'd like to classify the relationship as "it's complicated!" because what if the polygon is not a simple polygon and there are no dots in the middle of it as well? Nevertheless, I googled Pick up and found that...



Hmm, I have not tried his formula out but this calls for more Mathematical Investigation at another time!

Lecture #4.

The lecture started with Dr. Yeap "reading our minds" again. But alas, as with all the other quizzes and mind teases, there was another number pattern to discover in the very first activity for today. I tried this "trick" with my sisters aged 16 and 11. My 16 year old sister got the trick immediately while the 11 year old one took a little bit more time. I had a hand of practicing "differentiation" on the spot and had the older sister to find out more possibilities of obtaining the solution while I allowed the younger sister to figure out more by us having more number combinations. It was fun and we had a good time.

We worked on word problems tonight. The 3 different kinds we touched on tonight:


1. Change Situation: (Where there is a change involved. There is an initial amount and an unknown amount after the change takes place.)


E.g.: There are 37 cupcakes. Jane gave away 19. How many cupcakes are left?


This can take place with discreet quantities (e.g.: marbles, stickers, etc) and can also take place with continuous quantities (quantities that cannot be seen one by one, e.g: water, rice, etc and have a unit of measure, e.g.: $, kg, m, etc)


2. Part-whole Situation: 


E.g.: There are 37 children in a class. 19 are boys. How many girls are there? 


For word problems like this, children should be exposed to a different variation (discreet/ continuous) of subtraction/ addition problems. Variation is important, not repetition (Zoltan Dienes). Sets of the unknown should also be reversed. 


3. Compare Situation:


E.g: I have $37. I have $19 more than you. How much do you have? 


For word problems like this one, never introduce the keyword strategy to children where "more" means to add because it is not the case all the time for every word problem with the word "more" in it. 


We also worked with fractions and how to introduce them to children. I took home the following tonight:

• Fractions -  first taught as a part of a whole.
• For the initial introduction of fractions, do not write actual fractions out. Children have just learnt number representation and quantity and to understand 1/4 as a quarter or 1/2 as half would confuse them. 
• Instead write them as 1 fourth, 3 fourths, etc.
• When introducing addition of fractions, write out 1 fifth + 3 fifths = ? instead of 1/5 + 3/5 = ?
  Reason being that any child that understands the concept of 1 apple + 3 apples would understand the above problem as there is no change in concept, only the noun is different.
• Saying "1 out of 5 plus 3 out of 5" is taught much later.  
• Fractions - eventually taught as a part of a set/ quantity. E.g.: 1/2 of 35 children, 1/4 of $100, etc.

Last but not least, something fundamental that we should never tell children is that equal parts look the same. Tonight's class proved over and over that being equal does not mean that they have to look identical, nonidentical parts can be equal still! Since we are on the topic of fractions, I realize we are more than halfway through this module too!

Lecture #3.

We had a guest speaker tonight, Ms Peggy went through the meaning and purpose of "Lesson Study". Observing the case studies she provided for us through video recordings allowed me to identify factors of good math teaching and learning for numeracy development in Early Childhood.

 As a group, we were to form as many different structures as possible with 5 cubes. For our group we realized that when all of us formed structures together, we ended up with a few repeated structures that were the same hence not fully utilizing the cubes to the maximum. Therefore we ended up having someone being the "eye" to check for repeats while the other two of us continued to form structures. In the end as a class of different groups we had many structures that were the same but we also had structures that were different from the other groups. This shows that creativity never runs dry and that there are more than one ways in solving a problem, even approaching it. 

What stayed with me from this lecture is the idea of differentiation. Differentiation is the concept of carrying out the same lesson but catering to the different levels/ needs for each child in the class. I think that differentiation in a math lesson is a must, lest children get discouraged and find math too difficult a task/ subject. 

Mathematical Investigation is an excellent tool where differentiation can be carried out. Mathematical Investigation is:

• An activity that is divergent in nature
• Allows pupils to explore and experiment mathematical ideas and situations in many directions
• Allows pupils to demonstrate problem solving, thinking and communication skills

In Mathematical Investigation, students are assessed on their ability to:

• Describe
• Justify solutions
• Document steps they have taken to arrive at solutions

As teachers, our roles in Mathematical Investigation are to:

• Facilitate
• Observe
• Listen for children's reasoning
• Ask questions at the right moment to help further thinking
• Provide waiting time for children to think and discuss ideas

Lecture #2.

Second lecture tonight. If there's anything obvious that I am gleaning from this course already it definitely has to be that every day items can be used as tools in and for a math class! Buttons, cans, cards yesterday and today, pick up sticks. I never saw math being presented to me in the form of games and "magic tricks" but this is how we have been working on math these couple of days so far.

Today I realised that math has been completely and totally misrepresented to me in my own early childhood and school days. The top famous words that came together with math for me then were, "formula", "working", "memorise", "model drawing", "method". It made me think that the goal of math was to have you never fully understand what it really is. The imagery of having a classroom full of students perplexed with "memory work" and "formula" for the purpose of algebra, trigonometry and all that jest seemed illogical to me. Math was painted as a rigid and straight square with only a one way track of finding your answers to a particular question. I often gave up because the task to find that "one way track" was too obscure and hidden. Instead of answers I had more questions. "Why do I have to do it this way", "Why does the unknown have to be x"... I wanted to understand math and why it worked only a certain way but today I learnt that the broad goal of math at the end of the day actually is to provide life skills that are competent in the 21st century. There you have it - plain, nice and simple without complication.

It was an eureka moment for me to learn that ultimately, the goal of math is to create thinking and problem solving skills such as these:

• Communication
• Reasoning
• Justification
• Representation 
• Generalization
• Identifying patterns
• Making connections
• Making relationship 
• Development of number sense
• Visualization
• Meta-Cognition (the part of the brain that monitors thinking)
• Managing information 
• Overcoming weaknesses

See how Mathematics has been misrepresented all these while?

I also enjoyed the presentation of long division in a different way - the traditional way of 651 divided by 3 versus the other method where 651 was further broken down to 600, 30 and 21 to be divided by 3. Dr. Yeap presented the traditional long division method in a hilarious way ("This is how you do it, don't ask why...", "Just bring the number down, yes, because I say so...", "We are doing division but you must know your multiplication also", etc) and it brought back memories of the way of how I was taught to do long division in primary school too! I am so glad it is not taught the traditional method in school now, because it makes so much more sense and it allows children to think further beyond a set of given "dos" and "don'ts". It makes math come alive.

"We are not teaching Mathematics. We are teaching children." Dr. Yeap (2011). 

What stuck with me from this lecture is that when children learn math, this principle has to be observed.

• CPA - Concrete, Pictorial, Abstract by Jerome Bruner who was a constructionist, believing that children construct ideas by going through concrete details, pictorialization and then finally abstraction. 

Therefore when teaching children, teachers should never approach the abstract first as this would definitely be completely daunting towards children!

At the end of the day, Mathematics (and all subjects, for that matter) is a vehicle for thinking. It is not the destination. As teachers, we need to bear this in mind so children can in time realize it too for themselves and thus as a result, the broad goal of math will be accomplished.

Lecture #1.

We had our first lecture today and the very first activity got everyone thinking and working on numbers already. Throughout the first lecture, the many activities allowed me to approach math in a different manner. Instead of looking at them as a "problem" to "solve", I realised that what was waiting to be discovered were number patterns that would in return help to decode and unravel the solution. It was especially exciting trying to figure the pattern to the cards (lesson and activity 4). My partner and I worked backwards and started with card #9 and #8. We eventually got on until #5 but somehow our method couldn't work with #4 onwards... I still wonder why and would definitely want to figure that out still!

We also learnt the different uses of numbers in this session and how we as educators should present them to children in the right manner and at the suitable time for them to grasp and understand.

• Ordinal numbers - used to indicate position with respect to space and time. 
• Number patterning - a child's possible first number pattern is introduced in rote counting where there is an addition of 1 to the next number.
• Rational counting (cardinal numbers) - there are two forms, namely discreet and continuous to count quantities. In rational counting, units of measure are eventually introduced (around Primary 2) but not in the initial stage of counting.

I learnt through this lecture as well that the ten frame is a useful tool for a teacher to gain insight on where a child stands in terms of having "number facts". E.g: The ability to "see" from the frame that 10 and 2 make 12, 5 and 7 make 12, and even other patterns, etc.

These are a list of things that teachers can take note of in terms of a child's mathematical level of understanding and capability from a child's work and interaction on a ten frame.

• Counting all - child is be able to begin counting from 1. 
• Counting on - for example in the sum 3 + 4, child begins counting on from 3 (or even 4) and does not fall back to begin counting from 1. 
• Counting on with commutative property of addition
• Skip counting 
• Conservation of numbers
• Advanced emerging ability in place value

What stayed with me the most for this lecture is seen in the picture that follows.

In the English language, we have countable and uncountable nouns. In Mathematics, we also have countable and uncountable things. This picture is an example of a group of things that cannot be counted. Yes, we can still count how many items there are in total. However for a young child beginning to count, it would be a confusing task to accomplish. For children, counting should first be introduced with identical things (e.g. buttons of the same kind, same unit, same set). They can then be moved on to counting in identical sets before eventually moving on to the nonidentical things. This picture of things is therefore uncountable for young children.

Reflections for Chapter 2.

Exploring What it Means to Know and Do Mathematics


Knowing and doing math are two separate things. This chapter made me reflect on my own learning experiences as a child in the area of math. Quoting from this chapter, "mathematics is a science of concepts and processes that have a pattern of regularity and logical order. Finding and exploring this regularity of order, and then making sense of it, is what doing mathematics is all about." I like how math is being defined in the quote because it shows that beyond just a set of topics with problems to solve, it is a subject that involves investigation and allows us to makes sense. It is a subject that causes us to unlock thinking skills that are vital in our everyday life.

Sadly, I never saw math in this light growing up. Math lessons always seemed to follow the same predictable structure where the teacher would explain certain methods in doing certain sums, ask a few questions, have a few children try out sums in front of the whole class on the board and then we would be sent back to our seats to complete our math exercises alone and individually where any form of discussion could mean you were copying answers. It was not only a lonely journey of learning, it was also intimidating. Moreover it was anything but discovery-like or exploratory.

From this chapter I have understood the importance of a teacher to create a "spirit of inquiry, trust and expectation" in the math classroom. Math becomes an invitation, not just another piece of homework, task or assignment for children. This made me reflect on how to make that invitation plausible in the early childhood classroom and I realise that this invitation might not be accepted or readily taken if it remains only at a verbal level.

Reflections for Chapter 1.

Teaching Mathematics

The very first chapter of our given text has given me much insight on how Mathematics should be taught. I particularly like the 6 principles that are listed in this chapter. The authors also put it aptly to say that the excellence in mathematics education involves much more than listing content objectives. 

In my opinion I would think that these 6 principles given by NCTM (National Council of Teachers of Mathematics) can actually be paired nicely with our own 6 teaching and learning principles (or more commonly known as iTeach) as stated in the Kindergarten Curriculum Guide by MOE in 2008. Here's how I think they could work together:

1.     The Equity Principle from NCTM together with the principle of Holistic Development from iTeach.

The Equity Principle is all about having opportunities and support for children's mathematical learning in all aspects. In the same way, our curriculum guide also advocates for children to be provided opportunities for children’s development and the area of numeracy is also mentioned.

2.     The Curriculum Principle from NCTM together with the principle of Integrated Learning from iTeach.

Both principles state that mathematics/ learning experiences should be integrated into whole, interdisciplinary activities rather than isolated as a subject on its own.

3.     The Teaching Principle from NTCM together with the principle of Teachers as Supporters of Learning from iTeach.

Parallel to each other, these principles assert that teachers who understand what they are teaching as well as understand how children learn will be able to support and extend children’s learning and thinking.

4.     The Learning Principle from NTCM together with the principle of Ample Opportunities for Interaction from iTeach.

Both principles believe that a positive environment that is nurturing allows children to make mathematical/ learning conjectures. A positive environment for learning is one that gives children the space to interact with materials, peers and teachers.

5.     The Assessment Principle from NTCM together with the principle of Children as Active Learners from iTeach.

The Assessment Principle states that assessments “include ongoing observation and student interaction” and are not solely based on class tests or quizzes. In the same way,  iTeach also recognises that effective learning takes place when children are actively involved in meaningful tasks. The purpose of assessment is for the enhancement of children’s learning.

6.     The Technology Principle from NTCM together with the principle of Engaging Children in Learning through Play from iTeach.

The Kindergarten Curriculum Guide states that it is essential for “teachers to provide a senses-rich environment” that would facilitate learning. This principle goes hand in hand with the Technology Principle with the intent of providing the technology/ materials necessary that would enable a child to have increased exploration of the mathematical world.

Introductions.

Hi Professor Yeap (and all who stumble upon this blog)! I am Elizabeth Koh from BSc 05 and this site is set up for the module EDU 330, Elementary Mathematics.

I was thinking of a name for this blog and came up with "kinder-math". Whether you read it as "kinder", the short of "kindergarten" or "kinder", the comparative form of "kind", this blog aims to mean both.

Mathematics, arithmetic, and all things numeric never really went along well with me during my school days. I was always more literary than mathematical and numbers never became my friends the way words did. It was to my surprise when I found out that the course outline for this module required us to set up a blog for our mathematical reflections, and for starters it has debunked the idea within me that this module will consist purely of numbers and formulas. For one, we are starting the module here with words, phrases, sentences, reflections and 50 of them for that matter, taking into account my other course mates'...

In this module I hope to discover another side to Mathematics that perhaps I have overlooked during my school days. And in my pursuit, I believe I may just find that Math, after all, is kind! Despite my personal weakness in this area, I would also like to acquire knowledge and understanding as to how Math can be taught to little ones in a way that does not intimidate while challenging them to stretch their thinking muscles. I hope to be able to put what I have learnt into practical use to especially help children who find Math a challenge.