Georg Alexander Pick

The man who developed a simple formula for calculating area of a polygon on a linear grid.

Area = Interior points  –  Exterior points/2  –  1

In class however, we took a longer time to identify the pattern to find the area of the polygon drawn on the grid paper. But a classmate was able to come up with her own formula. Pretty impressive! We saw various polygons of different sizes and tried counting the square area. The polygon I drew was a little trickier but Dr. Yeap was able to help out using division of fractions. It was so much easier to calculate the area of a polygon that has all sides parallel to the grid points.

PS: Dr Yeap, nothing in my online research of Dr. Pick shows that he came up with this theorem while sitting in jail bored and alone. He was however, a math and physics genius who died in a concentration camp.   :p

This activity reminded me of how I introduce to the children the formula for finding area of a parallelogram. I do not provide the children with just the formula. Instead, we allow children to explore the cut-out shape of the parallelogram independently and discuss their findings first.

“First, we move the triangle shown on the left to the right. Then we have a rectangle. The area of the rectangle is the SAME as the area of the original parallelogram.”  🙂

Students can benefit significantly when using the CPA approach based on the work of Jerome Bruner whose name is oftentimes mentioned in Dr Yeap’s class. 🙂 With this in mind, we will use this approach for our upcoming assignments and lessons.

The concrete (manipulatives) stage consist of actual contact with objects. For example Cuisenaire, rods, base ten blocks, measuring tools, etc..

The pictorial stage helps students to bridge the gap between concrete and abstract with strategies such as drawing pictures, diagrams, graphs for students to read and interpret.

Abstract refers to the symbolic representation, writing numbers or letters to demonstrate their understanding of the task.


We also explored various word problems and identifying the problem structure used. The four basic structures for addition and subtraction story problems are:-

1. Join

2. Seperate

3. Part-part whole

4. Compare

In my 3rd grade class we introduce the various models to solve word problems. These models help to give meaning to the number sentences. The children will also get to see that addition and subtraction are connected.

For example, there are 12 children in the bus. 4 are boys. How many girls are there?

Children can solve it either by using the part-part whole model: _ + 4 = 12

or they can use the seperate model: 12 – 4 = _

It is important for children to see the relationship between the addition and subtraction facts. One way is to practice our basic facts and to be familiarize with the fact families.

The first question posed by Ms. Peggy on that session was, “What is lesson study?”.

In school, we do a word study unit with the third graders where we  investigate and understand patterns in words to replace the traditional spelling instructions. So my guess to Ms. Peggy’s question was,  lesson study is a process whereby teachers study a topic or unit and then plan a thorough lesson. I was close, however there’s more to lesson study than that.

Lesson study is a professional development for teachers. Unlike the usual workshops or conferences we attend , lesson study involves the teachers to work collaboratively and on a small number of “study lessons’. For example, the group of teachers in the video that she showed us identified a research theme that they want to explore. The teachers not only need to plan the lesson, they also have to pick one of the groupmember to teach and the rest will observe. In the video, we see Ms. Devi carrying out the lesson while the other educators watch on closely.

As a class we were asked to critique the lesson. We then categorized our observations into different aspects of good teachings. And for this lesson we came up with 9 different aspects of good teaching for a Math lesson.

1. Sitting Arrangement (Circle or semi-circle)

2. Level of engagement (How are children involved)

3. Use of materials/manipulations

4. Flow/Sequence of lesson

5. Classroom Management

6. Communication (Teacher-student, student-student)

7. Questioning techniques (No. of questions, types of questions)

8. Disposition

9. Differentiation

In my opinion, lesson study helps teachers to focus and explore the research theme better through observation and implementation. Yes, it might be challenging and nerve-wrecking for teachers to have their fellow colleagues observe them as they teach, but just like the video of Ms. Devi, I learned a lot on how I could improve myself as a facilitator of math.

Yesterday, I took some time off math class to show the children the trick that I learned from class on Tuesday.

Remember the Dice Game? I was curious to see if my eight year olds were able crack the code and look for the patterns just like the Primary 1 children did in the video.

I started out by telling them that I had a magic trick to show them. You should have seen the look on my children’s face when I call out the sum of the digits facing inwards. They were surprised and at the same time curious to know how I came up with the answer. Right away I saw some of my boys looking intently on the dice that I was holding and I could tell that they were doing some math in their head. After a few tries, I asked the children if they know how I came up with the sum. A few hands were raised and some thought it was an addition sum for the 2 sides shown and others thought i subtracted the two numbers.

After a few more guesses, I told them that I would write the number on the board so that maybe this will help them see the pattern better. Right away after writing my 3rd set of numbers, I have kids doing some counting either with their hands or mentally. I had 1 boy who shouted out the number 14. I told him to explain further, he said all three sets equals 14. There are three addends. Then I started hearing more excited chatters among the children. One by one, they began to understand how I derive to my answer.

Suddenly Jon, my youngest  3rd grader raised his hands and said he had another method. Well I was surprised, firstly because Jon rarely raises his hand in class and secondly, I didn’t know of any other methods. Eeek! 😐

Jon knows that the opposite face of a dice adds up to 7. So when I showed one face of the dice, he instantly knows the value of the dice facing inwards. Smart boy! Even Ms. Liana didn’t noticed that.

We had fun and it was a nice change to our usual math routine class. It was interesting for me to see which of my students were able to stick to the problem and try to solve them. Most of the children were able to apply the 5 key components of problem solving and thinking – Generalization (patterns, relationship, connections), Visualization, Communication (language, representation, reasoning, justification), Number Sense and Metacognition. I would definitely like to give more problem solving questions to the children and share the “tricks” we learned in class from you.

Had an interesting and comprehensive class last night with Dr. Yeap.

Firstly, those math problems that he introduced to us would have turn me away and I would have given up almost immediately. However, he guided us and made us envisage other methods we could use to solve the problems making me more determined to solve it. I find it very helpful to be working with my classmates and as a whole class. We get to listen to other people’s solutions and questions.  Who would have thought that there are so many different and interesting methods for just 1 question? And best of all, we derived to the same solutions.

One thing that I agree fully with Dr. Yeap is that teachers need to have a professional knowledge of what they are teaching. For example when we converse with parents or other educators, we should be able to use the appropriate vocabulary to explain to the point. Some vocabulary I learned from last night are; cardinal numbers, ordinal numbers, rational counting, norminal numbers, count on, conservation of numbers and commutative property. Other things I learned is the four pre-requisites to counting and the different strategies to add two single digit numbers  5 +7. Some of these strategies are to count all, count on, making 10 (ten serves as the basis of our number system) and another strategy I suggested was making doubles (7×2)-2.

Looking forward to tonight’s class. 🙂


Math. I would consider math my strongest subject among all the classes I have taken. In math, you either get the answer right or wrong. Simple as that, unlike the open ended questions you chance upon in Literature or a history paper.

There are two kinds of math people; those who have a deeper understanding and can do the math, and those who simply solve the problems without knowing the reasons why they did it a certain way. I used to be the latter, but hopefully now after five years of teaching the same math topics,  I’m better able to reason my answers and explain my thinking.

Currently, I am working as an instructional assistant in an international school. For the past five years, I have taught and tutored math to students. Even so,  I’m still in the learning process of understanding my math concepts and figuring out ways to make my lessons easy for the students . Asking myself time and again if my lesson makes sense or if I’m approaching their questions correctly and to the point. Most importantly, I have to stop myself from repeating the memorized procedure that I have learned back in school to my students. For example, 3 x 100= 300 (Write 3 and add the two zeros at the back). cringe*

Thankfully, the school has adopted the Everyday Mathematics (EDM) curriculum developed by the University of Chicago School Mathematics Project and it is based largely on the NCTM standards. They provide teachers with instructional activities to conduct the lessons and include differentiation options for supporting the needs of  all students. In our school, we are able to provide a wide range of support  from a Math Support class for students who need the extra help to a Gifted and Talented Education class to challenge the high level math students and allowing all children to grow as learners at their own rate.

What I like about this curriculum is that it reviews and supports previously taught concepts and skills. For example, if a lesson on fractions was introduced 2 units before, most likely there will be questions on fractions in their upcoming workbook pages or test. This helps students to maintain their skills and to help teachers identify students who will need more revision on the topic.

EDM also practices the five process standards of Problem Solving, Reasoning and Proof, Communication, Connection and Representation. These process standards refer to the mathematical process through which students should acquire and use mathematical knowledge.

Real Life Problem Solving: A lot of the lessons emphasizes the application of mathematics to real world situations. The lessons are linked to situations and contexts that are relevant to everyday lives.

There are also a lot of emphasis on Communication for students to explain and discuss their mathematical thinking, in their own words. There are questions found in the book that helps to support this process such as “Explain how you found your answer.”, “How do you know that your answer makes sense” and “Describe the pattern.” Opportunities to verbalize their thoughts and strategies give children the chance to clarify their thinking and gain insights from others.

Every day, there are certain things that each EDM lesson requires the student to do routinely.

~Math Messages are problems displayed on the screen for students to complete before the lesson and discuss as an opener to the main lesson.

~Mental Math and Reflexes are brief activities to help strengthen children’s number sense. Such activity includes basic math facts, estimation…

~Math Boxes are pages in their math workbook where children complete problems independently and most of the time this is where teachers assess the student’s knowledge on the topic based on the five content standards: Number and Operations, Algebra, Geometry, Measurement, Data Analysis and Probability. This could be a formative or summative assessments.

~Homelinks or homework is sent home everyday to reinforce the day’s lesson and to connect home to the work at school.

What Does It Mean to Do Mathematics?

It is important that the children are not only able to solve the math problems but also learn mathematics with deeper understanding and build connections with other mathematical ideas thus building new knowledge from prior knowledge. The children should be able to explain the procedure that leads to the solution and check to see if their answers make sense on their own.

I am looking forward to learn from you how I could improve myself as a math teacher that teaches through problem solving and not just from memorized math equations/drills. Most importantly, I want to be able to share the joy of doing math with the children and have them benefit from the next few classes I have with you.