Monday, December 2, 2013

BSc08 MAT150 Developing Problem-Solving Skills Session 5 on 2/12/13

Today we looked at how to make lessons problem-solving in nature as well as how to differentiate instruction for advanced learners as well as remedial students. We looked at the idea of tiered instruction as well.

This group even included me in the problems they created. Can you identify the other characters in the problems?

Why is this a problem?

What can you ask an advanced learner to do?

Monday, November 18, 2013

BSc08 Reading 1 Poster Sharing - Polya's Model

The class was asked to make a poster to share with fellow educators. Some of the students made a poster to share their understanding about Polya's Model and the kind of questions suitable at each stage.

I encourage fellow educators to have a poster of Polya's Model - Let's Understand, Let's Plan, Let's Carry Out and Let's Look Back in our classrooms ...and make it suitable for the age group.

Sunday, November 17, 2013

MAT150 Developing Problem-Solving Skills Session 3 on 18/11/2013

Don't forget we meet this evening and tomorrow to discuss instructional strategies and also linking mathematics to other domains.

After these two days, the assigned reading is Chapter 4 - Planning in the Problem-Based Classroom.

Based on your reading, compile five photographs to remind colleagues in early childhood sector what is important in planning for exciting problem-solving lessons for young children. Please bring these five photographs (can be simply printed on your own printer) with annotation (limited to three sentences) for the next class meeting in December. You can use the guiding questions at the end of Chapter 4 to guide you.

An example will be shown here later.

Today we look at instructional strategies and assessment methods.

We solve Problem 31 - How Many Pieces?

Along the way we reviewed important bits about teaching beyond ten, choice of materials for shapes, and the language of measurements.

We also saw the creation of the English Language as a problem-solving process.

Problem 32 - How Many Coins in the Can?

We discussed some differentiated instruction strategies via a discussion on a lesson to compare length of paper strips.

Several challenging topics were brought up.

Sunday, November 3, 2013

Development of Numeracy in Early Childhood Education

This course, customized by Marshall Cavendish Institute for Ministry of Education Singapore, reviewed the question - how children learn mathematical ideas.

Dr Yeap Ban Har's email is


Manipulative Materials

Classroom Warehouse Products website

EAI Education's website

The (clothes) number pegs etc are available in many neighbourhood stores and I saw some in Popular as well.

The 'eggs' are from a shop behind Ang Mo Kio Market (Ang Mo Kio Avenue 10) ... I will get the exact address on my next visit. Watch this space.

The giant playing cards are about $3 a pack from 'pasar malam' type stores.

The music videos are from Debbie and Friends

My favourites include Jack and the Beanstalk, Cinderella and Goldilocks. Actually I like all the Debbie and Friends videos!

Thursday, October 10, 2013

MAT150 Developing Problem Solving Skills 10 October 2013

We reviewed problem solving using this video

Yi Yang (age 1.5 year) pretending to solve a maths problem
We solve a problem connected to polygons with area 5 square units and ended up observing a pattern.

We also solved a problem connected to packing 19 beans in bags of 2 or 3 beans as well as two other related problems - 10 bicycles and tricycles have 28 wheels and the possible values of 10 coins of two types - 5-cent and 20-cent coins.

We reviews the CAT to DOG problem by trying to do TOAD to FROG (how about STEPSISTER to CINDERELLA?)

We looked at various heuristics - some concrete, others pictorial and yet others abstract.

Five core competencies - visualization, number sense, looking for patterns, metacognition and communication.

If you are absent or missed the discussion please submit a solution to the problem with one main point that you can discuss based on the solving of the problem.

Reading to be done before November's sessions - Chapter 3 and complete on of the tasks by presenting it in the form of an A4 poster for sharing with colleagues. Submit this when you come back in November.

Wednesday, October 9, 2013

MAT150 Developing Problem-Solving Skills 9 October 2013

Introduction - Problem Solving and Singapore Curriculum

Experiencing and Discussing
Show two halves.

Polya Four-Stage Model was discussed.
1. Understand
2. Plan
3. Do
4. Look Back

Problem solving includes exploring, specialising and generalising. (Mason)

Problem solving include self checking.

We completed three other tasks to understand what is problem solving.

In three steps, we changed CAT into DOG.
Only one letter can be changed each time.
Each word must be an English word.

John and Mary agreed to meet at 6 o'clock at a restaurant. John's watch is fast by 2 minutes. Mary's watch is slow by 5 minutes. John arrived at the restaurant, looked at his watch and said, " I am 5 minutes early." Mary did the same and said, "Oh no! I am 5 minutes late."

How long does one person have to wait for the other?

We also looked at Mrs Lim's happy problem (PSLE Preliminary Examination Problem from a school).

Mrs Lim usually spent $290 on several bottles of vitamins at Healthway Pharmacy. During a storewide discount of 20%, Mrs Lim found that she could buy 2 additional bottles of vitamins with the same amount of money.

Were you able to make the connection that the 20% discount is the cost of the two additional bottles?

If you are absent or missed the discussion please submit a solution to the problem with one main point that you can discuss based on the solving of the problem.

Reading to be done before November's sessions - Chapter 3 and complete on of the tasks by presenting it in the form of an A4 poster for sharing with colleagues. Submit this when you come back in November.

More photos on my Facebook.

Thursday, September 26, 2013

EDU330 Elementary Mathematics Session 4 on 26/09/13

Do you see multiplication in this arrangement of lanterns for the Mid Autumn Festival in Chinatown in Singapore?

Problem 13  is based on a 'mind-reading' read. Choose two  digits A and B.
Make a 2-digit number AB and add the numbers A and B to get a sum.
Find the difference between AB and the sum.
If you know A you can figure out what the difference is.

In Problem 14 we explored ways to subtract 3 1/4 - 1/2

In Problem 15, the three little pigs shared 4 pizzas equally.

Problem 16 is to figure out the multiplication facts for 7 by the doubling method, the distributive property and the subtraction method.

Problem 17
Draw a polygon on geoboard paper. How many times is it as large as the square?
Jessica shares her solutions at her blog.

EDU330 Elementary Mathematics Session 3 on 25/09/13

We started to look at basic concepts in fractions. Problem 11 Share this equally among 4 persons. The class were challenged to find a way where the four equal parts are all different shapes. This is one solution.

Problem 12 is on sharing 3 fourths of a cake equally among three persons.

Monday, September 23, 2013

EDU330 Elementary Mathematics Session 2 on 24/09/13

EDU330 Art at KKU from Jimmy Keng

If you are absent on any day / part of the day, please make up by solving the problems discussed that day.

Problem 5
Based on As I Was Going to St Ives, show how many persons, cats and sacks there are in the rhyme.

Problem 6
This is for two players. Select any number less than 20. Take turns to subtract either 1 or 2. The winner is the player who gets to zero. Think about the strategy to win the game.

Problem 7
Think about different ways to calculate 5 + 7 + 6.

Shareefah shared a video on ten frames on her blog.

Problem 8 Jack & The Beanstalk (My Favourite Video!)
Share 51 golden eggs among three persons.
In how many of these ways do Jack get more than the other two persons (each person gets at least one egg)? What if they are shared equally among three persons?

EDU330 Elementary Mathematics Session 1 on 23/09/13

Notes from Session 1

Sample Quiz Item is included.

If you are absent for any session, please solve the problems discussed in class and submit them together with your Final Assignment. Today's problems are well captured by Lydia.

Not at the Casino

More photographs at Facebook.

Problem 1 - Which letter is counted 99?
This is an example of a problem that has to be presented orally.

Problem 2 - Arrange the ten number cards.
This is an example of a problem that has to be acted out. There is a video on the card trick by Nur Shahirah.

Problem 3 - Textbook Problem
One machine shreds one truckload of paper in 4 hours. Another shreds one truckload of paper in 2 hours. How long does it take for both machines to shred one truckload of paper when they are working at the same time?

Problem 4 - Make a rectangle using tangram pieces.

Vygotsky in Action

Sunday, September 22, 2013

EDU330 Elementary Mathematics BSc09 Welcome

The course starts this coming week.

Your blog links have been added. Otherwise please give me the blog addresses in class and I will add them.

I encourage you to read Eve Tham's entry - several points that she brought up will be discussed in tonight's session which focuses on what to teach and how to teach them in early years.

Thursday, August 22, 2013

Reading Some of Your Blogs

The Card Trick
Today I read several of your blogs. I like the videos on Chujun's. There was one video on geoboard where the narrator said she made an equilateral triangle. Question: Is that possible? Prove it.

Her blog also gives us a glimpse into some theories we discussed in class.

Harlina's entries are interesting - it gives the hard theories in a light-hearted way. Plus the tangram book she unearthed for us may be of some interest to all. Given such interesting read, the three entries do not seem enough to for my pedagogical appetite.

Kelly's blog gives you a clear glimpse into what we did in class. She also included some things she found on the internet. The geoboard video shows a range of problems using the geoboard - by fixing the number of dots inside, number of dots on the perimeter, area and son on. Watch it.

What is really helpful is when the blogger gives readers an idea what to expect in each video. That way I do not need to spend time watching it if that is not what I am looking for. What is also helpful is the concise summary that some of you gave (it helps those who were not in class have a glimpse into what happened). I really enjoyed reading the extensions that some of you make after each lesson.

Friday, August 16, 2013

EDU330 Elementary Mathematics 08/17/13 Session 6

Today, you had fun with Ms Peggy Foo on lesson study and solving a bunch of enrichment lessons as well as to learn about differentiated instruction. From the work you produced, I noticed you have enjoyed yourselves and were thoroughly engaged.

Today, I read Salmah's blog as well as Marina's blog - Marina has a a talent in picking relevant videos to share with colleagues. I found the videos to be the most useful thing in her blog. Salmah's approach of using a question to focus each blog entry is helpful to parents seeking specific answers. So if my child cannot count a set of things, her entry Child Unable to Count ? might be helpful.

One approach to blog to communicate with colleagues and parents is to select a relevant video and you provide a synopsis plus your professional views on the topic discussed in the video.

Another approach is to write blog entries based on frequently asked questions.

EDU330 Elementary Mathematics 08/16/13 Session 5

Quiz 2 - Write a journal entry for each.

Journal 5
Write two story problems to show the difference between 1/2 divided by 4 and 1/2 divided by 1/4.

In case you want more problems to solve, here is another one ...

B                              first line
AR ....

Using your own name, which letter in your name appears at the end of 99th line.

Thursday, August 15, 2013

EDU330 Elementary Mathematics 15/8/13 Session 4

Today we review whole numbers and fractions as we move on to geometry and measurements.

We review Lesson 10 on 3/4 divided by 1/8 and then we moved on to a series of lessons on word problems involving the four basic operations.

Lesson 11
What is 2/3 divided by 5?

Lesson 12
1/5 of 2 thirds and 2/3 of 1 fifth

Lesson 13
Let's make pentominoes

Lesson 14
Draw a 4-sided figure with one dot inside it.
How large is it?

Wednesday, August 14, 2013

EDU330 Elementary Mathematics 8/14/13 Session 3

1. Review Whole Numbers
- Lesson 7 How Many Monsters?
- Lesson 8 Dice Problem

2. Quiz 1

3. Break - please discuss with your group members the plan for this Friday Museum Activity

4. Fractions
- Revisit Lesson 3 Four Equal Parts
- Lesson 9 Subtracting Fractions
- Lesson 10 Multiplying Fractions

Monday, August 12, 2013

EDU330 Elementary Mathematics 8/13/13 Session 2

"No matter how lucidly and patiently teachers explain to their students, they cannot understand for their students."

Lesson 4 is based on Jack & The Beanstalk

5 + 6 + 7
Think of different ways to find the total of the three numbers.

Method 1 is to make ten. 
How many different ways to make ten?

Lesson 5 
Jack and his two friends share 51 beans equally.
Think of different ways to do this.

We discussed how this leads to long division algorithm. 

We had a discussion based on As I Was Going to St Ives
The song goes all the way back to 1800's and was found among ancient Egyptian artifact - see History

Lesson 6
AB x C = DE 
What are the digits A, B, C, D and E if they are all different?
None of them is 0.

We saw the four ways whole numbers are used and did problems based on the four operations (Lesson 5 - adding, Lesson 5 - dividing and Lesson 6 - multiplying).

We added metacognition and number sense to patterns and visualization as big ideas in mathematics learning.

EDU330 Elementary Mathematics Sample Quiz Item

Photograph by a student in Pathlight School

Other than counting objects in a set, we can also count number of ways to arrange objects in a row.

This is one way to arrange the four houses in a row.

Find the number of different ways to arrange the four houses in a row. Show your method clearly.

Students do not learn permutation (counting ways of arrangements) until in their teenage years.

Describe one experience that 5-year olds can make sense of that provides a foundation for the learning of permutation.

EDU330 Elementary Mathematics 8/12/13 Session 1

P/S - if your name is still not listed on the side bar, please resend me the address of your blog and if you are absent for part or all of any lesson, please complete the problems discussed in your absence and submit these with your other journal entries on friday or saturday.

Creating a Mathematical Climate in the Classroom
1. How Children Learn Mathematics?
> We discussed two perspectives including the CPA Approach (Bruner)
2. What are the Big Ideas in Mathematics?
> We discussed looking for patterns (Lesson 2) and visualization (Lessons 1 and 3),

1. Introduction - What's new about school mathematics in Singapore and the US?

2. Lesson 1 - Tangram "Make a rectangle."
The class surfaced three ways of helping children learn - through exploration, through teacher modelling and through teacher scaffolding.

Literature Connection - Grandfather Tang's Story

3. Break

4. Lesson 2 - Name
The letters in my name is counted in a certain way (ask a friend to show you if you missed this one).
Which letter is counted 99th? How about your own name?

5. Course Briefing

6. Lesson 3 - Equal Parts
Make four equal parts.

7. Journal Writing

8. End of Session 1

Photographs are available on my Facebook
One of the students did this for a 7-piece solution

Saturday, July 27, 2013

EDU330 Elementary Mathematics - An Introduction


Watch this video.
Doing Maths involves a Lot of Discussion

We start in August, after the Hari Raya and National Day festivities.

You are expected to read Chapter 1 and Chapter 2 and make a blog entry in the form of a note to parents to show that you have understood the readings and have an opinion on the readings. 

The purpose of the note is to help parents understand that mathematics their children learn may not be the same as what they themselves experienced as a child (probably about two decades ago, at least).

The blog entry should not be excessively long (typically fewer than 300 words) and, hopefully, is creative and interesting to read. You can use an existing blog or start a new one. 

Photographs, videos and so on may be used to make a point. Please create a blog for this and daily reflection and send the address of your blog to before 12 August 2013. 

Your name should appear on the side bar of once your blog address has been received.

Friday, July 26, 2013

MAT150 Developing Problem-Solving Skills Session 6

Math Trail at Bolles School
Examples of activities that can include problem-solving tasks are maths trail (see photograph), making art project (such as animals) using recycled / reused materials as well as games.

In the photograph, teachers of early grades were engaged in a maths trail task that required them to find the number of cubes needed to fill the empty water fountain.

The slides includes stimulus for problems-solving activities.

Once a teacher had a problem - she did not know which part of the rectangle is referred to as 'length'. Problems are not limited to mathematical ones. In class, we discussed examples of real-life and real-world problems. (in response to her problem, I would say the length of the longer side and the length of the shorter side but when we are lazy we simply say 'length' - longer side - and 'breadth' - shorter side.)

Slide 2 - The use of children's books to design problem-solving activities. we previously discussed The Doorbell Rang. Here a page from The Tortoise and The Hare is used. Can counting be problematic?

Slide 3 - The people who built this vertical garden needed to know how many pots of plants are needed. They had to solve a problem.

Slide 4 - Can this African painting inspire a problem? The scarves come in three colours, so do the blouse and the skirt. Here, we see six different outfits. How different outfits are possible?

Slide 5 - This artefact inspired a patterning problems among some teachers in the class. Can you use this to design a problem for young children?

Slide 6 - An ancient game (British Museum) from Mesopotamia. Is playing a game such as this one problem solving?

Slide 7 - Stone Rosette (British Museum) from 3300 B.C. The making of such rosettes is both an problem as well as an exercise. Explain.

Photographs of the course are available on my Facebook.

Thursday, July 25, 2013

MAT150 Developing Problem Solving Skills Session 4

The Doorbell Rang is a story for children that can be used for problem solving.

Children can be asked to figure out how many cookies each child gets. They can also be asked how many children can share the 12 cookies equally. They can also be asked to guess how many cookies on Grandma's tray (like guessing the number of easter eggs in a jar).

Monday, July 22, 2013

MAT150 Developing Problem-Solving Skills Session 1

Photographs of the whiteboard here.

What is Problem Solving? | Tasks that are novel or complex are often problems.
When there is no ready procedure for a person to complete a task, the task is likely to be a problem to the person.

Saturday, April 6, 2013

EDU330 Communication Board

The course has ended but the learning continues.

Class Discussion on Subtraction of Fractions With Renaming
Please update you reflections on your blog. It is best done soon after the session. It is also good to complete your journal which will be submitted with your Final Assignment.

Discussion of your responses to the Quiz will be done here over the next one week.

Most of you managed to relate 3/4 divided 2 to a '(equal) sharing situation'. For example, this is a word problem that can be solved by doing 3/4 divided by 2.

Mother Jane bought a pizza.
She gave 3 fourths (of the pizza) to her two children to share equally between themselves. How much of the pizza did each child get? 
                                (Yan Ting, with minor editing)

Friday, April 5, 2013

EDU330 Data Handling & Differentiated Instruction

Today is the last session and the class baked cookies for me that come with a word problem.

EDU330 Algebraic Thinking

We solve two problems today - investigating polygons made on geoboard and arranging five consecutive numbers in a certain arrangement.

I recall doing this lesson with a group of gifted students in The Netherlands.

I particularly enjoyed students' attempt at finding the area of their polygons. In this photograph, two boys were quite animated in trying to calculate the area of the triangle.

Thursday, April 4, 2013

EDU330 Measurements & Geometry

Look at Piaget's idea of assimilation and accommodation.

Make-Up Work for Absentees - Find out by talking to your classmates and / or reading up and make short notes on:
(1) Explain 1/2 divided by 3/8
(2) First Three Levels of van Hiele Theory

Also complete the various problems solved in class today.

Today, I took sometime to read Ambiga's Blog and I like the link to different types of numbers. I also like the snapshots she took to give me an insight what she and her group members were up to when solving the problems.

Wednesday, April 3, 2013

EDU330 Fractions


* Review

* How Many Paper Clips Are There?

* Problem 6 - Equal Parts

Show 4 equal parts.

* Problem 7 - Present

Adrian and Bala shared the total cost of a present. Adrian paid $15 more than 3/8 of the cost of the present. Bala paid $25. How much did the present cost?.

* Problem 8 - Same or Different?

Show 3 fourths. Show 6 eighths.

Quiz 1

Write different stories to illustrate different meanings of 12 - 4 and 12 : 4 .

Explain three different ways to find the value of 12 - 7 .

Make-Up Work for Absentees

Make short notes for the most fundamental idea about fractions and how to teach equivalent fractions.

Also complete the Quiz.

Tuesday, April 2, 2013

EDU330 Situations Related to Addition and Subtraction

Photographs of the Discussion Board are available at my Facebook

Two types of addition stories were written by students in the course - part-whole and before-after situations.

For today's session, you might blog about different subtraction, multiplication or division stories. Or think about different ways to do 14 - 9 ...

Monday, April 1, 2013

EDU330 Different Uses of Numbers

Finishing a Race
In As I Was Going to St Ives, our friend added
7 + 49 + 343 + 2401 to get 2800. Counting the man, she got 2801!

In using numbers as cardinal numbers, our friend is counting wives, sacks, as well as cats and kittens. While it is fine to count the cats and kittens, it is kind of strange to count persons and sacks together!

In today's session we look at numbers used as cardinal, ordinal (space and time), measurement and nominal numbers.

We spoke about the difference between rote counting and rational counting.

We solved Problem 1 (Name Problem) and Problem 2 (Spelling Number Names).

Make-Up Work for Absentee(s) - Find out by talking to your classmates and / or reading up and make short notes on:
(1) Different Uses of Numbers with Examples
(2) Learning Theories by Vygotsky (social learning), Bruner (representations) and Skemp (understandings)
Please submit make-up work, which is typically one or two page long, with your Final Assignment, stating the date of absence.


How can you tell the number of eggs in this photograph?

Tim, a K2 boy, is unable to tell you the correct number of eggs in the photograph.
What information can you provide Tim's parents with?

Saturday, March 23, 2013

EDU330 Wheelock College / SEED Institute BSc 07: Welcome

This course is Elementary Mathematics where the focus is for pre-school teachers learn a combination of mathematics and its pedagogy with implications for early childhood settings.
Kindergarten Lesson at Seoul Foreign School

This course for the cohort BSc 07 starts on 1 April 2013. Students must have received Course Information from SEED Institute.There is a pre-course reading that has to be  completed and you need to have a blog set up to document your learning.

Students are expected to read Chapter 1 and Chapter 2 of the textbook and make a blog entry to show that they have understood the readings and have an opinion on the readings. 

Textbook: Van de Walle, J. Elementary and middle school mathematics: Teaching developmentally (8th Edition). New York: Longman. ISBN: 9780132879040

How many of each type of animals are there? What are the skills that a teacher needs to help child develop before this task is given to the child? If a child is unable to do this task, what is the cause? This and other questions will be explored in this course.

Source: O R Tambo International Airport (South Africa)