Monday, August 19, 2013

Session 5 - 16 August 2013

Children learn through..

1) Visualisation
2) Matacognition
3) Generalisation/ Patterns
4) Number senses
5) Communication

Activities for children to learn maths:

Teaching Math Without Words, A Visual Approach to learning Math from MIN...

Session 4 - 15/8/13

Pick's Theorem

Figure 1
Figure 1

A lattice point in the plane is any point that has integer coordinates. Let P be a polygon in the plane whose vertices have integer coordinates. Then the area of P can be determined just by counting the lattice points on the interior and boundary of the polygon! In fact, the area is given by
Area(P) = i + (b/2) - 1
where i is the number interior lattice points, and b is the number of boundary lattice points.
Presentation Suggestions:
A neat application of Pick's Theorem is that you cannot draw an equilateral triangle whose vertices lie on the lattice points.

Ask the students to try, making the point that the base does not need to be horizontal. Some convincing approximations may be obtained, but explain that none of them is really equilateral. This is true because the area of an equilateral triangle of base A is an irrational multiple of A2. If it is drawn on the lattice, then A2 is an integer ---use the Pythagorean theorem--- and the area is irrational, contradicting Pick's theorem. (That's also a good opportunity to prove that the square root of 3 is irrational.)
The Math Behind the Fact:
Pick's theorem is non-trivial to prove. Start by showing the theorem is true when there are no lattice points on the interior.

**Do you agree?

Wednesday, August 14, 2013

Session 3 - 14 August 2013



I had learnt that I need to say 3 tenths instead of 3 over 10 or 3 upon 10 etc. Learning about the different ways which we can solve fractions and not our old usual way where we have to memorize the formula and I remembered during my schooling days, we were told to remember formulas and practice and practice. I always don't understand why on earth I have to remember formulas and I am definitely not good in memorizing things. Just like what Dr Yeap said, Howard Gardner's multiple intelligences does not include the part, memorization intelligence. If I had a chance to ask my teacher then, I would certainly asked and understand why it was done that way during our time.

The Dice Problem

We were given 2 die and need to find out the total hidden numbers from the way we see the die. It was interesting that opposite sides of the dice makes the total value of 7 and there are the different methods and ways to solve the hidden numbers. And it all goes back to the basics when we see a problem and wonder how to solve a problem, we:

1. Generalize the pattern
2. Use visualization
3. Number sense
4. Metacognition

I also learnt that numbers must always go with a unit and what are some of the reasons why children could not count. And as educators, we need to find out the underlying issue or problem that the child if facing instead of ignoring it and think hopefully one day the child will be able to catch up. Providing variation of activities are also important as it will help us understand and have a progressive plan for the children. When we have too many similar or same activities, the children will not be able to build their knowledge but instead, staying at the same spot with no progression.

Professionals develop techniques to allows them to find out the root of the problem. That's for sure.

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.  ~S. Gudder

Day 2 - Whole Numbers

If you ask Google what is whole number?

Whole number is...

A whole number is a number that is not a fraction of a number, a percentage or has a decimal. If you have a number like 21.32, it has a whole number portion (21), but in itself, this number is not a whole number because it contains a fraction (.32). The term integer or natural number may be used to define a whole number.

Today we learnt about whole numbers and using the ten frame, we did and find different ways to find the total of 5, 6 and 7. It is very useful as it led us to see other kinds of patterns, like arranging to make tens, counting in 5, or arranging it to see it in 6 each which makes all in the total of 18. And that is one of the way which the children learn, through visualization. The use of the concrete materials are also important in visualization and another way which we learnt how children learn is CPA (Concrete, Pictorial, Abstract).

We also learnt about the 4 uses of numbers; Ordinal, Nominal, Measurement and Cardinal. And in ordinal numbers, there are different concepts in time and space where we should use it appropriately, for example in a race where only the 1st is distinct, we can say that that person is 1st in the race, however, for the others whose ranking are still not determined, we can only say, that person seems to be 2nd from the finishing line. And that is the distinct and difference between time and space.

So far, I have learnt and understood:

How children learn:
1. Patterns
2. Visualization
3. Number sense
4. Metacognition

Ways to teach a child:
1. Exploring
2. Role-Modelling
3. Scaffolding
4. CPA (Concrete, Pictorial, Abstract)

Introductory to Elementary Mathematics - Day 1

Since my secondary school days, I have a fear of Mathematics. I can't seem to get the correct answer or correct solution to the questions and a part was also because of the teacher I had back then. Whenever I or the others asked a question, all she does was to shut us off and to ask us to get back to what we were doing. And since that episode, no one  asked her any questions and I simply rely on tuition.

This module, I tell myself, to have an open mind and to unlearn and relearn Mathematics again or as much as I can in order to be able to teach at least, my own children. The first lesson was making a rectangle and who knows How interesting it can be by using tangrams and using limited pieces to try and form a rectangle. It came to a point where we became more challenged and tried more and more pieces and its simply amazing to How when humans have a sense of achievement and then willing to further challenged themselves to the next level. I particularly like what Dr Yeap mentioned, "No matter how much you want to forget but can't, it is knowledge." Like the smell of durian, it is simple words yet carry much impact. And this type of learning is through visualization whereby we see how the tangrams can form a rectangle. Then we move on to another interesting lesson which is using Dr Yeap's name and find out the 99th letter in it. There are many ways we can do of course but I guess ultimately, it is to let us know that we can find patterns and that is one of the way that children learn.

We also learnt about the ways to teach a child, through exploring, scaffolding and role-modelling. It is close to what we learnt in our theories and what we put in practice for our method in teaching young children. And if we put in the same theories as what we used in Mathematics rather than seeing it as only practise, memorization for Mathematics, the children could definitely do better and show interest in the subject.

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.  ~John Louis von Neumann

If two wrongs don't make a right, try three.  ~Author Unknown

Saturday, August 10, 2013

Note to Parents

When we talk about Mathematics, numbers, symbols, equations, sums are what we often think about and know about. It is challenging for some, interesting for some and of course, easy for some. Mathematics is simple more than completing sets of exercises or mimicking processes, but how we generate solutions for problem solving, checking if it makes sense and it can help to connect to what we see in the real world.

The six principles from the Principles and Standards for School Mathematics are:

1. The Equity Principle
  • Students must be given the opportunity and adequate support to learn mathematics regardless of personal characteristics, background, or physical challenges.
2. The Curriculum Principle
  • Students must see that mathematics is an integrated whole, not a collection of isolated bits and pieces.
3. The Teaching Principle
  • Understanding what the students know and need to learn and then challenge and support them to learn it well.
  • Need to understand the content of mathematics and select meaningful instructional tasks.
4. The Learning Principle
  • Learning Mathematics with understanding and also the ability to think and reason Mathematics.
  • Students are required to evaluate their own ideas and make mathematical conjectures and test them.
5. The Assessment Principle
  • Having ongoing assessment and observations are essential as it can help in making instructional decisions and gathering data of students' understanding where the teachers can use to better make daily decisions and support their learning.
6. The Technology Principle
  • Technology tools are essential for doing and learning mathematics and permits students to focus on mathematical ideas and to solve problems in ways that are often impossible without using technology tools.
To become the teacher of Mathematics, the following are important points to look out for:
  • Knowledge of Mathematics
  • Persistence
  • Positive Attitude
  • Readiness for Change
  • Reflective Disposition
Mathematics requires effort and its important that the students, families and the community acknowledge and honor the effort in learning Mathematics and to work together to help the student problem solve and understand the concepts of Mathematics.