Day 6 - Patterning and differentiation

We know that doing math is also looking out for patterns (generalizing).

3 steps to facilitate this are:
- What do you see? (Describe, observe, visual)
- What do you think? (Relationship, connections, any patterns?)
- What do you wonder? (Reasons and infer, from my prior knowledge and the discovery that I saw)

I make relation to these 3 steps to provide experiences from simple to complex and concrete to abstract. Just like when we did this activity.

When we examined the patterns closely, we made assumptions and then inferred to think of the next possible pattern.
Then we got to see that actually there's a relation between the series.

1st figure: A square with a square hole in the middle
2nd figure: A cross with 4 corners
3rd figure: Square with 4 square holes (A combination of the 1st and 2nd figure)
4th figure: make up of 4 numbers of the 2nd figure.
From these patterns, we concluded that the 5th figure will have 16 square holes on the square paper.

Besides the CPA approach, we can also design activities using differentiation instructions. Children who are more advance, they should be challenge with problems that they are more likely to be capable of solving. At the same time, I can focus on children who needed more help.

Differentiation instructions by:
- content (the objectives)
- process (different level of difficulties)
- product (end product)

To end the last blog entry for this module...

The high-light of the day The card trick! It's not magic but trial and error!

Day 5 - calculating angles through CPA

Visualization is the word for today!

I agreed that the competency to visualize needs to be nurtured from young. How to train the children to visualize?

1. Exploring and playing with materials (e.g. playing tangram)

2. Having opportunities to handle and manipulate materials of different textures.

3. Before teaching the concept, allow time to play and explore first. Allowing children to discover something related to the concept before being taught.


Two problems were done in class. We were given the task to find the angles given.

 

Using the concrete approach makes the result more visible to me.

When it comes to visualizing, I needed more time to analyse.

The way we were taught math from young were really structured. It was through rote learning. Now I could see how learning can be effective when teaching starts with exploration of the concept first. With that we see why that certain formula works for the concept. Just like why the 3 angles in a triangle formed 180 degree when added together.

The concrete method: When I cut the 3 corners of the triangle and then place them together, they formed a straight line, which is 180 degree (like a half circle)!

Last but not least, the five important points to remember when teaching and learning math are, Generalization, Visualization, Metacognition, Number Sense and Communication (When we talk and discuss the problem during the process and also to explain how we came up with the solutions)!

Day 4 - Geometry Fun!

Concrete, Pictorial, Abstract

Let’s continue to see how the CPA approach is being put into practice when teaching and learning math...

1. Whenever introducing or teaching basic shapes (square, triangle, circle and rectangle), there is a need to provide concrete shapes for the children to touch and manipulate.

2. Children can also learn to measure the area of shapes using geoboard!

I have always felt that learning to measure area is tedious. After today's session, I have learned a new way to find out the area of different shapes formed on the geoboard.

I have geoboard in my classroom and the way of manipulating this material in my class was to form different shapes or pictures using rubber bands on it.

I can further challenge the children (the advance children) to count the area of the shapes that they formed by providing a simple square as a unit for them to count with. Just like what we did in class.  

Form any shapes with a dot in it and then count the number of square units needed to form that shape.  

 

Create different shapes and figure out the area. Try this! Have fun playing on the Geoboard!


Geory Pick came up with the Pick's Theorem. He tried out and compared different shapes. He then created a formula for measuring the area of different shapes.

 

Day 3 - Fraction

Fraction can be simple and not as complicated as I have learnt during my primary school years back in the 90s.

If the parts are the same, we can name them. For example, half, one third, one quarter, one fifths.
Teaching fraction to young children through dramatising game


The correct way of saying fraction - 3 tenths
not 3 over 10
not 3 upon 10

Uses of fraction:
- fraction as part of 1 whole thing (e.g. a pizza)
- fraction as part of a set. (e.g. a group of children)

One of the "oh" moment that I got from the day was how we can do division of fraction in the way that I have never thought of that it will work. Just like this!

 

Day 2 - Whole Numbers

The concept of whole numbers is the most fundamental concept in mathematics.

1.       From a story, it leads to a mathematics problem. For example, a video story “Jack and the beanstalk” was shown and it led to counting with the actual things (the beans).

 

* A simple video that illustrate how children can learn math through visual*

 Gum balls counting

2.       Ten Frames: As a teacher who teaches mathematics using this simple tool, I see lots of learning opportunities for the children.

-          It’s a tool which children can see numbers visually (aid in one to one correspondence).

-          Knowing the number bonds of 1 – 10 (e.g. 9 is make up of 4 & 5)

-          Number conservation: No matter how we move or place the beans in any box in the frame, it’s still the same number of beans.

-          Knowing place value. When all the 10 boxes were filled up with 1 bean each, it’s a complete set (tens). Each bean will represent ones.

-          These leads the children to learn that there are other ways to make tens and to find the total when counting.

3.       Division is not that complicated if we use number bond

4.       Cannot count 2 different nouns. For example, 2 cups + 4 bowls =? They are 2 different things, so we cannot add them, unless we change them to the same thing. For example, 2 cups + 4 cups = 6 cups.

5.       There are different kinds of numbers

- Ordinal numbers: Numbers for time and space (1st, 2nd, 3rd)

- Cardinal numbers: Numbers that tell the amount for countable objects

Doing math is looking for patterns, visualization, having number sense and metacognition. If any of it is lacking in the children, the teacher can model and scaffold. Don’t have to explain, they will gradually find out while exploring and manipulating the materials.

Day 1 Journal

After such a long time away from doing mathematics and besides introducing mathematics activities to the children in my class, today's session really made my brain work! The experiences provided made me want to work on it to find out the solutions.
Firstly was the tangram! It's fun to manipulate and move the shapes to form rectangles. Through this, I see that math can be open ended enough to have more than a way to solve.  

Next, using Dr Yeap's name to find out which letter is counted 99th. It's really cool to see the patterns as I counted. There can be different ways to solve it. The very first one that I thought of was using division. I then moved on to count one by one. That's where I saw the repeated patterns and I stopped to count through the pattern! Look!

The third activity helps me to see how children can move from concrete to pictorial to abstract when learning math. First I manipulated the paper to fold, then to see if the lines formed any equal parts (pictorial). After which I needed to know if they were equal abstractly, but I can find out the equal parts concretely when I cut and overlap the pieces to see.

So, math learning is looking for patterns and visualization.  It's built on from our prior knowledge and with opportunities for hands on too!

Teaching and learning Mathematics

Notes to Parents

There are many influential factors that play a part in how mathematics are taught or learnt. In addition, to be effective in teaching mathematics, we need to have the mathematics knowledge and also knowing how learners learn.

There are six principles involved in teaching mathematics. These principles allow us to know that it is not just the concept that learners need to know when learning mathematics. Furthermore, there are five process standards to show how learners should practice and gain mathematical knowledge. The principles and standards illustrate how experiences should be for learners and how it means to understand mathematics.

Doing or learning mathematics is not by memory like what we always thought of. It is by understanding the pattern. Doing mathematics is being able to generate strategies to problem solve, applying the strategies to see if it works and also if the answers make sense. “Productive struggle” helps learners to learn mathematics. It is not leaving the learner with a question or problem without equipping him or her with prior knowledge. It is to allow them to try different approaches to find the solution and not directly intervene to help when the learners got stuck. This act is supported by the six principles and five process standards. Knowing the patterns and understanding the concepts, that is doing mathematics!

There are two learning theories which complement to how learners do mathematics. The constructivism theory explains how learners can construct new knowledge upon prior knowledge. Just like building blocks, having a foundation base in order to add on more.

Sociocultural theory explains how learners can be supported by peers or people who are more knowledgeable to reach their zone of proximal development (ZPD). In this way, learning mathematics will be productive!