## Sunday, 7 April 2013

Make up lesson for Thursday, 4th April 2013

How do we solve 1/2 : 3/8?

We need to do cross multiplication to solve the fraction problem and derive the answer.

1/2 : 3/8 = 1/2 x 8/3 = 1/1 x 4/3 = 4/3 =  1 1/3

To learn and understand geometry, we need to know their content goal which applies to all grade levels.  These consists of  shapes and properties, transformation, location and visualisation.
Van Hiele explains these properties of geometry through his 5 stages of Geometric Thought.

Stage 0 - Visualisation

This is the first stage. Students at these stage name the shape of objects based on the visual characteristics they observe. Anything with 4 sides would be a square and 3 sides is a triangle. Students begin to classify shapes according to how they look.

Level 1: Analysis

Students begin to realise that geometrical shapes have special properties that make them different from each other. They begin to notice that a square has 4 equal sides while rectangles are made up of four sides, opposite sides parallel, opposite sides same length, 4 right angles, congruent diagonals.
As students use the different shapes, their understanding of the properties of shapes increase and get refined as they begin to look at symmetry, perpendicular and parallel lines and use them to represent classes of shapes.

Level  2: Informal Deduction

Students begin to look for similarities in shapes and develop a relationship between the shapes based on their common properties. E.g. a shape with 4 right angles is a rectangle and it can also be a square because a square also has 4 right angles. They begin to use logical reasoning when they look at shapes and classify them.

For myself, I have always enjoyed "playing" with the various levels of Tangrams we have in the centre for the children to play. I have found them challenging to the mind as I place the different shapes to create a basic shape. Gratification comes when all the funny shapes are able to create one complete shape.

Monday, 1 April 2013

I was apprehensive as I came to the first math class today. I was wondering what I will be learning. The first activity was a pleasant surprise. It was so easy to find the answer. After counting 3 rounds, I had the answer to Dr Yeap's name. The 99th position was at letter 'N' of 'BANHAR'.  I was even able to see the pattern and explain it.

However, I felt tortured by my long name. I couldn't find the 99th position. I had to write each number until 99 and check a few times before I found the answer. Then, after finding the answer, it was another challenge to look for the pattern. It was only after the discussion that I was able to realise that there are different patterns. I had to do some subtraction of numbers to find the pattern.

This learning was later linked to Vygotsky’s theory of social constructivism where children learn best in a group setting and Jerome Brunner who had said that learning begins with concrete to abstract.

Dr Yeap talked about the CPA approach which mentions that children should begin learning from exploring concrete ideas/materials and then progress to pictorial before being exposed to abstracts concepts.

Today we also talked about the different types of numbers - cardinal, ordinal, rational, nominal, odd, even, cute, perfect and prime numbers. When counting, children need to use similar things to be able to achieve their goal of counting. It is difficult for young children to perceive when the nouns are different.

I also realised that that I have been guilty of committing similar mistakes through my unawareness at times. I will have to consciously rectify this.

Tuesday 2,  2nd April 2013

When children have conceptual understanding, they are able to make connections to new ideas to what is already established in their mind. Teachers are the best persons to help them in this task.

What is big and what is small? How do you define it? Is the big circle drawn by Jane also big to Tom. What is more/less, large/small. All these are qualitative measurement. How would children be able to differentiate the terms to match the other persons perception? They need to be taught in comparison of similar items. E.g. o or o  for circle. Teachers need to plan the lesson carefully with attention paid to specifics. The teacher needs to plan the learning outcome of the lesson.

When counting, teacher will need to be able to know how the children count and arrive at their answer, through counting or subitizing. We subitize when we are familiar with a particular subject.  We can tell the total number just by looking at the dots on a dice or cookies in a jar without counting.  Children may also need a benchmark to subitize when they try it. To be able to do it correctly, children should be given similar materials/ containers to compare.

We did addition problems today. It was fun creating the different stories. Language plays an important role in creating understanding through problem sums.  Children need to have their learning scaffolded to a higher level of thinking through enrichment activities.

Wednesday 3, 3rd April 2013

Humpty Dumpty sat on the wall
Humpty Dumpty had a great fall
All the king's horses and all the king's men
Couldn't put Humpty Dumpty together again!

How can we use this as a stimulus to teach children?
• how high was the wall
• what height should the egg fall before it will crack
• counting the horses vs the men - can we classify them? how?
• how many horses and how many men were there
Maths does not always have to have numerical beginning or deal with only numbers. Literacy approach can also be used to teach mathematics concepts and stimulate children to a higher level of thinking. In early childhood maths, topics are not listed, as such, they learn through visual literacy.

We looked at fractions in a fun and visually stimulating way. We folded paper in different ways to get equal parts.

We were able to fold into rectangles and squares. There were some who also managed to fold it into  equal triangles. it was fun trying to fold it in different ways.

This was one of the CPA approach we had been introduced to in lesson 1.

Friday, 5th April 2013

Today we looked at dots in a box and used 4 dots to form different geometrical shapes. It was interesting because we were able use differentiated ways to find a polygon, such that, there are 4 dots on the side.

As we created the polygon, we were able to create shapes that were double the size of another similar figure. We learned how to measure the polygons to get their areas. Different shapes with different angles were created. This method of doing it on paper is for older children.
(The younger pre-schoolers can be introduced to simply creating polygons using geo boards with elastic bands.)

Today we were also introduced to Gauss, a mathematician who lived from 1777 to 1855. We learned how to add numbers across, just like Sudoko, to get the sum of three numbers in both directions. This helps to develop intelligence, encourage visualisation and patterning concepts in children.

Saturday, 6th April 2013

When the same 2 digit numbers are used to do subtraction by reversing the numbers, they will give a standard figure. E.g.  98 - 89 = 9,  54-45=9, 32-23=9

Children are taught to do subtraction by renaming.

There are 3 questions that the teachers needs to ask when teaching values:
1. What do I want the students to learn?
2. How do I know if they have learned or not learned?
3. What if they can already do ?
These questions can be answered by the 4 stages of teaching:
1. model to teach by modelling good teaching
2. scaffold
3. provide opportunities to practice and do
4. explain only as a last level of teaching as it requires high level process to understand
Graphing can also be introduced to students using familiar things that they can relate with. We did a graphing activity on the age groups of the students in our class.

We were able to see the number of students in each age group by just a glance. This information was then used to create graph.

## Chapter 2: Exploring What It Means to Know and Do Mathematics

It was interesting to read that "mathematics is the science of concepts and processes that have a pattern of regularity and logical order." I had never thought of mathematics in scientific terms before this.

The use of language when doing mathematics, creates an awareness of higher level thinking in children as they use conceptual verbs to problem solve and derive strategies and answers. Through the use of explicit language, students are able to make connections and see patterns in a problem as they engage in productive struggles - "students must have the tools and prior knowledge to solve a problem, and not be given a problem that is out of reach, or they will struggle without being productive; yet students should not be given tasks that are straightforward and easy, or they will not be struggling with mathematical ideas" (Walle, Karp, Bay-Williams, 2013, p.15)

Doing mathematics involves the science of pattern and order to understand and apply concepts. Once students are able to see these patterns, they are able to make connections and engage in their productive struggle. Besides this, using technology, such as a calculator makes counting accessible for students who can't count yet. Making predictions, drawing pictures or even using objects to solve problems are some basic ways to do mathematics.

Mathematics theories are rooted in the constructivism and sociocultural theories of Piaget and Vygotsky.  As theorized by Piaget, "Integrated networks,  or cognitive schemas are both the product of constructing knowledge and the tools with which additional new knowledge can be constructed."
(Walle, Karp, Bay-Williams, 2013, p.19)
According to Vygotsky's sociocultural theory, , semiotic mediation is used in classroom interactions through the use of diagrams, symbols, representations to make meaning which falls within the ZPD (Zone of Proximal Development) of the learner.

Therefore, the mathematics teacher needs to:
• build new knowledge from prior knowledge
• provide opportunities to talk about mathematics
• build in opportunities for reflective thought
• encourage multiple approaches
• engage students in productive struggles
• treat errors as opportunities for learning
• scaffold new content
• honour diversity
Different types of tools and manipulative can be used to represent concepts and test knowledge and understanding. These include websites, geo boards, base-ten blocks, spinners, number lines, unifix cubes, counters)

Students should also be encouraged to think aloud hypothetically when they have solved a problem. Getting them to  recall and explain how the problem was solved would present them with opportunities to recall and build upon their existing knowledge. When the students are able to understand and make sense, they also feel confident in their ability to understand and do mathematics which will increase their performance level.

Chapter 1: Teaching Mathematics in the 21st Century

"In this changing world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their future." NCTM (2000, p. 50)

It is important for teachers to be able to focus on mathematical thinking and reasoning as they prepare to help students learn mathematics. When teachers understand and apply the Principles and Standards for School Mathematics in the school curriculum by integrating it with other subjects, they provide a rich and meaningful mathematics education for the children.

Mathematics is not just about knowing and calculating numbers, but  learning it "with understanding, actively building new knowledge from experience and prior knowledge." NCTM (2000, p. 20)
It requires children to base their learning on the Five Process Standards which include:
• Problem solving
• Reasoning and proof
• Communication
• Connections
• Representation
As the teacher uses these processes in her mathematics classroom, there is coherence and comprehension taking place. Children learn to draw knowledge from prior experiences and make connections to new ideas through reasoning and communications as they problem solve.

Teachers who understand the pedagogy behind teaching mathematics to young children, strive to create an environment that will offer and encourage all children with equal opportunities to learn. this can be achieved through the use of different media, such as technology, reasoning, communications, and reflections. As children interact with the different media available, they become exposed to using strategies to derive answers rather than focus on only "one right answer".