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.