Tuesday, November 1, 2011

Questions at AMSTI Math Leadership Symposium

Do primary school teachers in Singapore teach more than one subject?

Singapore primary teachers are trained to teach more than one subject (two or three subjects). See details at National Institute of Education. Typically, a teacher teaches more than one subjects, especially at lower grade levels. In some cases, especially at higher grade levels some teachers may teach multiple classes of the same subject. Teachers teaching Mother Tongue, Music, Art and Physical Education are more likely to be teaching only one subject. It is only at secondary level that teachers specialize somewhat (two teaching subjects).

Other than English and Mathematics which are core subjects other subjects include Science, Mother Tongue Language as well as non-examinable subjects like Art, Music, P.E., Social Studies, Health Education, Civics and Moral Education and a host of other programs such as Porgram for Active Learning (PAL).

More to come

Sunday, October 30, 2011

Textbooks in JC

I am a student self-studying mathematics from Singapore's textbook at the 10th to 11th (Secondary 5) grade levels. When I finish these, I would like to continue following Singapore's curriculum. Which books (series) do you recommend. I plan to study advanced topics in depth, like statistics and calculus? I would like to follow the newest books, ones followed currently in Singapore by JC students.

Student in USA

In JC, students do not use a textbook because the lecturers provide their own lecture notes, like in the university. If you have not done Additional Mathematics, you may work on this - it is a book that includes trigonometry, calculus and other advanced topics. This is used in Grade 9 - 10 by advanced students. It is available from the same place you got your other books.

Wednesday, October 19, 2011


I hope you still remember me. Last time I contacted you regarding congruence and similarity. Now, my son is studying in Grade 8.

If you don't mind can you clear my confusion regarding one problem of percentage. I have difference of opinion with the mathematics teacher and I want to clear my concept. The problem is as follows:

When we are converting 2/5 to percentage we write as follows:

2/5 x 100% = 40%. ---------- (1)

The teacher says it is not necassary to write symbol(%) with 100.

2/5 x 100 = 40%. ------------(2)

You will notice that % is missing from (2). Can you please explain is it also the correct practice not to write the symbol % with 100. My view is that 2/5 x 100 will result in 40 and not 40%.

I am afraid that this mistake will results in marks being deducted in the IGCSE.

A father in Saudi Arabia

You are right 2/5 is equal to 2/ 5 x 1 and 1 = 100/100 which is written as 100%.

2/5 x 100 = 40 as you said. 40% is equal to 0.4 (not 40).

Perhaps the teacher knows that, in a lenient way or marking, candidates get full credit whether they did (1) or (2). But I am sure you rather your son learn what is mathematically correct rather what is minimally acceptable to earn a credit in the examination. Please advise your son to write the mathematically correct sentence. I also trust that you will help him understand why (2) is not correct (although it may still earn a credit in the examinations, according to the teacher).

Wednesday, October 5, 2011

From a Dutch Homeschooling Parent

I'm a homeschooling mother from the Netherlands and I have used (and still use) Singapore Math for my children in all the years of their primary education. I can't rave enough about this method of math education: the books are great, the bar diagrams are marvellous - I wish I had learned mathematics this way in my time.

Now my oldest child is starting Secondary Education the upcoming year. In the Netherlands secondary education consists of 6 years and the level of math at the final exams is relatively high (higher than say, American High School level - to give comparison). However, instead of preparing my children for their final exams with Dutch secondary math books, I prefer to keep on working with Singapore Math. Even though I can't really compare Dutch and Singapore High School Math, I very much like the way Singapore Math has build a strong math education in my children in such a thorough and painless way.

But in my search for information I got a little confused about all the available series. There seem to be:

1) New Mathematics Counts Series
2) New Elementary Maths Series and
3) New Syllabus Mathematics

Would you be able to explain to me the difference between these series or can you give me an advice on what to use for the High School years of my children? I'm aware this is not the type of problem question you usually receive on your blog, but I very much hope you can answer me just the same.

For the primary years we're using the My Pals are Here Maths Series (Grades 1-6), and I'd like use the best available SM high school sequence. I prefer to use the series that is used in most Singapore and/or International High Schools. For example: I understand that for the primary years My Pals are Here Maths and Shaping Maths are the most common used series in Singapore and International schools: MPAH in 80% of the schools, Shaping Maths in 20% - roughly estimated.

Do you happen to know the ratio of the three above mentioned series for the secondary years (New Mathematics Counts Series / New Elementary Maths Series / New Syllabus Mathematics)? Or do you know any distinctive features?

It seems that all Singapore Math distributors, either in the US, UK or Singapore are all pushing the series that offers them the most profit, so I don't know who to believe in that area. Buying all the books for four children is quite an investment, so it would be a pity if I invested in the wrong series. I was even on the verge of abandoning Singapore Math for a curriculum called The Art of Problem Solving (http://www.artofproblemsolving.com/Store/curriculum.php) because I didn't have any clue of the right series. But then I found your blog, I hope you can find the time to answer me.

Thanks in advance for your effort.

A Parent in The Netherlands

In Singapore, some students study 4 years and others 5 years for their secondary education. They then move to another two years in junior college (some opt to go to a polytechnic instead). The first four years leads to GCE O Levels (Grades 7-10) and the last two years leads to GCE A Levels (Grades 11-12).

You are right - there are many textbook series for secondary levels. In fact, more than the three you listed. Essentially they are all the same. Internationally, most schools / parents use New Syllabus Mathematics (NSM) and New Elementary Mathematics (NEM) simply because these has been around for many years. NSM is still sued in Singapore schools. NEM is no longer used - the publisher sometimes did not submit them for review or update it to make it a 100% fit with topics in the revised curriculum. All our textbooks must be reviewed by Ministry of Education. In my opinion, NEM is of good quality too.

New Mathematics Counts (NMC) is designed for academically weaker students - the program is to be done over five, instead of four years.

NSM is designed for students who have a strong foundation in mathematics. That is why other than the four books, there is a fifth book that advanced students opt for in Grade 9/10 (they use this book over two year to supplement the main text). This is Additional Mathematics. Many kids in Singapore study this subject. By the time they complete Additional Mathematics, they would have done basic calculus. Generally, if your child do well using the Singapore textbooks, he/she should be ready for ay kind of test.

In brief, NMC if the child is struggling with math. NEM or NSM if the student is quite good in math. NSM has the option for advanced topics.

See http://atl.moe.gov.sg/ for textbooks used in Singapore. Titles with Sec 5N are designed for students who tend to struggle somewhat with mathematics - the topics coverage is the same but done over five instead of four years. Books listed as NT (Nornal Technical) are for students who are moving to vocational course after Grade 10.

Bar Models

This morning I taught Primary 4 students word problems using bar models. I expected the students to get a clear picture by it, but actually not. They were so impatient to compute directly, even without reading the question :( They just depended on my instruction whether to multiply or to divide. So, I'm thinking of an activity at the beginning to introduce the bar model to them - perhaps by making the bars using color papers.

Teacher in Indonesia

This is not unexpected if the students are already used to a computational approach. All they want to do is to compute. Without a clear understanding of the problem, they will not be able to identify the correct operations.

Bar modelling begins in kindergarten when a teacher models the story using the real things and later pictures of the real thing. Later, 5 unifix cubes / snap cubes are used to represent, say 5 sweets. By Primary 2, they begin to use a bar to represent quantities.

Your idea of using paper strips is excellent. I do it all the time.

You can also give problems without numbers.

Johan has more sweets than Siti.
How many sweets does Siti have?

After students show the correct bars for number of sweets Johan has (longer) and number of sweets Siti has (shorter), tell them Johan has 6 sweets more than Siti. Ask them to put in the new information on the bar models. Finally tell them that Johan has 14 sweets. Get them to solve the problem.

Hope this helps.

Wednesday, September 14, 2011

I have attached a picture of a P3 Maths Question. The answer in blue is the student's answer.

My first thought was that the boy had given the correct answer. However, another parent pointed out that the question could mean having 8 sets of 2 square tables.

We would appreciate your view on this.

A Singapore Parent

I think the boy is correct. The test item would have been clearer if it is phrased this way:

A long table is formed by placing square tables side-by-side. Each side of the square table seats one person.

(This is followed by diagrams of two square tables, three square table and four square tables placed side-by-side.)

How many persons can the long table seat when 8 square tables are placed side-by-side?

I can understand why some adults may think that the answer is 34. They are thinking (because of the diagram) that a table is formed using two square tables and take the question to be asking for the situations when 8 such two-unit tables are used. This issue can be resolved by making the task clearer. One way to do so is suggested.

Saturday, September 3, 2011

Graduate Research on Singapore Math

I am a Prep teacher in a school in the Philippines. Our school has adapted Singapore Math for the past four years. I am currently doing a paper on Singapore Math at a university in Manila.

I wish to ask you if there are any materials, readings, or websites that you can recommend?


Ms. Patricia

Dear Patricia

Actually Singapore Math is based on learning theories. If you are doing a paper on Singapore Math you should be reading Bruner's theories on representations and spiral curriculum, also on Piaget's ideas on how children learn, Dienes theory of variability, Skemp's ideas on relational and instrumental understanding, and show how Singapore Math is consistent with learning theories.

If you are doing empirical research, there is a large number of areas you can research on e.g. how concrete materials help Prep children progress to pictorial and symbolic representations.

I will be happy to read your final paper.

Monday, August 29, 2011

Currently, I am teaching the topic on time.

We are beginning to teach pupils how to draw timeline instead of writing 2.30 pm + 30 mins = 3 p.m. I have reminded my pupils why 2.30 pm + 30 mins = 3 p.m. because a specific time cannot be added to duration. Despite going through in detail in class, I was surprised that some of my pupils were still writing the number sentence 2.30 p.m. + 30 mins = 3 p.m. though they drew the number line as well.

Moreover, their number line were not very helpful. It is obvious that that they had worked out the answers mentally in their heads.

Why is my pupils still writing it incorrectly?

Primary Three Teacher

This could be due to some of them being given extra (incorrect) instruction at home. The fact that the incorrect number number sentence give them the correct answer does not help your cause either.

It is good you tell them it is wrong. But be prepared for some who will take a longer time to represent the ideas correctly. Continue with the time line representation.

I am glad you pointed out that a point in time (e.g. 2 p.m.) cannot be added to time duration (e.g. 2 hours). Some students are taught by adults who make this mistakes themselves.


A few colleagues and I realise that our pupils have difficulty comparing fractions such as 2-sixths and 2-tenths. We provided alot of visuals such as fraction chart but the pupils are still confused. In My Pals Are Here! Primary 2, pupils are required to compare fractions such as 2 sixths, 4 sixths vs 2-sixths and 2-tenths.

They are confused between the concepts. Some can compare 2 sixths ad 4 sixths very well but not being able to compare 2-sixths and 2-tenths.

While others confuse the 2 methods involving the two tasks and end up with 2 sixths is bigger than 4 sixths while 2-sixths is smaller than 2-tenths.

How can I help our pupils overcome it?

Teacher in Singapore

If students are making mistakes such as 2 sixths is larger than 4 sixths then they are not making connections to visual representations. You should always use visual representations in early stages of learning fractions.

Early instructions in fractions is critical. Students should be well taught throughthe use of manipulatives and visuals that when 1 is divided into equal parts, the parts are named according to the number of parts. When 1 is divided into six equal parts the parts are each called a sixth, for example.

This allows them to reason the relative sizes of say 1 tenth and 1 sixth.

Thursday, June 23, 2011

Singapore Method

Estimado Dr. Yeap Ban Har: Desde la Región de La Araucanía en el sur de Chile, reciba un cordial saludo. Soy profesora de matemática y me gustaría saber los orígenes del Método Singapur para la enseñanza de las matemáticas, cuál es su filosofía, que está a la base del método, aquí en Chile sólo nos han capacitado en la forma y no más allá, necesito mayor sustento teórico para poder empoderarme y transmitirlo de la misma forma a mis estudiantes. Agradeceria me informara acerca de lo que me inquieta.

Mirhna, a mathematics teacher in Chile

The Singapore mathematics curriculum was introduced in 1992. One of its features is the CPA Approach which is based on Jerome Bruner's idea of representations. Bar model is used extensively in Singapore textbooks. The curriculum was developed in the late 1980s based on research and writings from around the world especially the US and UK. Our teachers are taught learning theories by Bruner, Skemp, and Dienes as well as ideas by Polya.

Wednesday, June 8, 2011

Durian Puffs

Here is a Primary problem which is from Primary 1 SA1 Paper 2 2011 from an unknown school. (Note: The mother who asked the question has since written back to say that it was not from an SA but from one of the continual assessment tools the school uses as part of its holistic assessment - it is called semestral review in this school. Also that she has mistakenly mention it is Paper 2.)

Mother has baked some cream puffs and durian puffs. She wants to put 8 puffs into a box. In how many ways can she put the puffs in order to have at least one of each kind of puffs in the box?

Is this a problem that can be solved by the bar 'model' method or some other way? What is its test objective?

How to solve by the 'model' method, or whatever method? Sorry, but I find this problem at Primary 1 really very tough, leh!

A Mother in Singapore

I am really not sure if you have got it right but this may not be a Primary 1 problem for these reasons:
(1) Schools generally no longer conducts SA1 at Primary 1 - that is the MOE guideline. Fornon-Singapore readers, SA1 is a semestral assessment after half a school year. It tends to be a written examination. MOE Singapore has suggested that children entering the first year of formal schooling should not be subjected to such assessment. Alternative assessment modes which may includes 'small' test at the end of units may be used.
(2) I have never heard of any school that has Paper 1 and Paper 2 in Primary 1. Paper 1 and paper 2 format tends to be for upper primary (P5 and P6) with Paper 2 allowing the use of calculators.

Note: It has since been established that it is a task used as part of a continual assessment that the school used. The person who asked the question has also clarified that she has mistakenly mentioned that it was from Paper 2.)

But it can be a Primary 1 problem because the content is from Chapter 2 (Number Bonds).

One way to solve the problem is to make a list - 1 cream puff + 7 durian puffs, 2 + 6, 3 + 5, 4 + 4, 5 + 3, 6 + 2, 7 + 1 (0 + 8 and 8 + 0 are out. You know why.). Thus there are 7 ways.

Bar model is not suitable. Rememmber that there are many ways to solve problems and model is only one such method. The objective of this item is to assess ability to solve an unusual problem.

Monday, May 30, 2011

From an Indonesian Student

I am a Junior High School student in Jakarta. I am 12 years old now. I like solving maths word problems. I have a maths word problem that I cannot solve by myself.

In a housing estate there are 1000 couples.
2 / 3 of the husbands who are taller than their wives are also heavier.
3 / 4 of the husbands who are heavier than their wives are also taller.
If there are 120 wives who are taller and heavier than their husbands, how many husbands are taller than their wives ?

I think the solution is 1000 - 120 = 880 husbands who are taller than their wives.
However, I am confused by the second and the third sentences in the word problem.

Made, 12-year old student in Indonesia

Yeap Ban Har writes: Let's start by assuming that a couple is made up of a husband and a wife. You may want to try to make a table (see photo - to be attached soon)

Also wife taller and heavier than husband means the same as husband shorter and lighter than wife.

Let's assume a husband is either heavier than or lighter than. It is possible that they have the same weight (mass) but let's not deal with that.

Can you continue?
(Note: Made has since replied that he was able to continue and solved the problem. See Comments for another suggested solution.)

Anyone would like to offer other solutions?

Monday, March 7, 2011

About a Primary 5 Problem

I came across this P5 question:

There are altogether 405 boys and girls. Each boy is given 35 sweets, each girl 23 sweets. In total, boys have 255 more sweets than girls. How many boys are there?

I can solve using algebra and trial-and-error, but the students (my nephew) was taught to apply this 'formula':

If it were all girls: 405 X 23 = 9315 sweets
Add the 255 difference: 9315 + 255 = 9570
Each girl and boy has 23 + 35 = 58 sweets
No. of boys = 9570 / 58 = 165

I can see that mathematically it works, but what kind of method is this, and how should this be taught to the students (by applying the formula blindly?) This was never taught to us during our school days???


I hope this is not taught as a formula! This formula is not general enough to warrant students learning it. there are so many problems and it is not productive to teach kids formula for each problem type.

I would not really if they were all girls. It is more like suppose everyone boys and girls each was given 23 sweets first.

There are altogether 405 boys and girls. Each boy is given 35 sweets, each girl 23 sweets. In total, boys have 255 more sweets than girls. How many boys are there?

If you use an algebraic method, you might let b = number of boys and set up this equation: 35b - 23(405 - b) = 255 which simplifies to 35b - 23 x 405 + 25b = 255 or 58b = 23 x 405 + 255 and hence b = [23 x 405 + 255] / 58. This is exactly the method that the teacher taught the class.

Question: How do we explain the algebraic solution? Why 23 x 405? Why plus 255? Finally why divide by 58 to get the number of boys?

Dividing a number by 58 to get the number of boys suggest that each boy got 58 sweets. But each received only 35 according to the problem! Why?

This method is actually one that many high-achieving students use and it is actually quite interesting and, at least to me, impressive that an 11-year old or 12-year old is capable of.

What is done is to give each child, boys and girls, 23 sweets. Thus 23 x 405 sweets are given out. Each girl got 23. Each boy also got 23. The 255 comes from the number of sweets the boys got (when each got 35) minus the number of sweets the girls got (when each got 23). So 23 x 405 + 255 tells us 23 x 405 (the number given to every child when each got 23) + 35b (the number the boys got when each got 35) - the number the girls got (when each got 23). The result gives number of sweets that the boys got (23, at first) + 35.


This method is quite clever in that if pretends to give each boy 23 first before the 35 that the problem requires. It also explains the algebraic solution that most adults are familiar with.

Tuesday, February 22, 2011

Lesson Study and Project Approach

What is the difference between Lesson Study and Project Approach (Lilian Katz)?

Kindergarten Educator in Singapore

"A project is an in-depth investigation of a topic worth learning more about. The investigation is usually undertaken by a small group of children within a class, sometimes by a whole class, and occasionally by an individual child. The key feature of a project is that it is a research effort deliberately focused on finding answers to questions about a topic posed either by the children, the teacher, or the teacher working with the children. The goal of a project is to learn more about the topic rather than to seek right answers to questions posed by the teacher."

This is taken from http://ceep.crc.uiuc.edu/eecearchive/digests/1994/lk-pro94.html

For example, students may learn about founding fathers in class. Some of them (or the whole class or one child) may become interested to learn more about Lee Kuan Yew. They may do internet research or read relevant books appropriate for their age or ask their parents questions about Lee Kuan Yew. They may pose questions that they can ask Mr Lee should they actually get to meet Lee Kuan Yew. This is an example of the Porject Approach that many eary childhood educators are familiar with - thanks to Katz. It is a teaching and learning strategy.

Lesson Study is a professional development activity where teachers 'study' lessons by discussing a lesson plan, by observing students and talking about what they see and so on.

As you can see it is possible to do Lesson Study on different teaching and learning strategies including the Project Approach.

Saturday, February 19, 2011

Engaging Students During Problem Solving

I was wondering...what are some ways that teachers in Singapore engage students while teaching problem solving?

Educator in the US

I understand that you teach math at college level? A good way to engage students while teaching problem solving is to ask students to suggest their way of solving the problem. Thus, students will see solutions of various degree of sophistication and choose one that is appropriate for themselves. For more difficult solutions, teachers can scaffold the process by asking questions and giving hints. The choice of problem is important - it must cater to a range of students. Focus on the process and not the final answer. In planning the lesson, anticipate how the students will respond.

See Marshall Cavendish Institute Facebook or http://singaporelessonstudy.blogspot.com/ for an example used with junior high school (grade nine)students in Japan.

Sunday, January 9, 2011

Problem-Solving Approach

What is a problem-solving approach?

This presentation made in Penang provides answer to this question.

Friday, January 7, 2011

About Mastery of Basic Facts


I have written a post on a blog regarding fact fluency and have received feedback that I am mistaken in that students in Singapore learn their basic facts to mastery...meaning with speed and automaticity. It was suggested to me that in fact, students in Singapore don't focus on facts and recalling them automatically is not an important focus of the program. True?

A teacher in the US

you are right that in singapore students are expected to learn their basic facts (1+1 to 9+9 and 1x1 to 9x9) to mastery.

true - that is not the focus of the program. the focus is problem solving and thinking - visualization, number sense and patterning. but that not not mean they are not expected to master basic facts.

the only thing teachers may want to note is that they have a fairly long time to do it.

students master addition facts in grade one, starting with number bonds and then they get lots of practice because they kept on using these in subsequent chapters.

for multiplication facts, they get a concrete and pictorial meaning making of multiplication before they are taught simpler basic facts such as multiplication of 2, 5 and 10 (and also 3).

plus, they get to learn how to figure out 3 x 6 from 2 x 6. also, 7 x 6 from 2 x 6 and 5 x 6 or 9 x 7 from 10 x 7. see any singapore textbooks - look for the dot diagrams (primary mathematics, my pals are here, shaping maths, math in focus, pensar sin limites - all have these dot diagrams)

all these are done over two years - grade two and three.
subsequently they get plenty of practice in larger number multiplication.

hope this helps