askbanhar

Questions at AMSTI Math Leadership Symposium

2011-11-01T14:06:00.000-07:00

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

Textbooks in JC

2011-10-30T00:33:00.000-07:00

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.

Percent

2011-10-19T06:30:00.000-07:00

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).

From a Dutch Homeschooling Parent

2011-10-05T00:56:00.001-07:00

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

2011-10-05T00:44:00.001-07:00

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.

2011-09-14T18:04:00.000-07:00

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.

Graduate Research on Singapore Math

2011-09-03T00:49:00.000-07:00

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?

Best,

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.

2011-08-29T19:47:00.000-07:00

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.

Fractions

2011-08-29T19:32:00.000-07:00

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.

Singapore Method

2011-06-23T07:54:00.000-07:00

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.

Durian Puffs

2011-06-08T07:41:00.000-07:00

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.)

Question:
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.

From an Indonesian Student

2011-05-30T23:05:00.000-07:00

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?

About a Primary 5 Problem

2011-03-07T05:46:00.000-08:00

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???

Chuck

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.

See?

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.

Lesson Study and Project Approach

2011-02-22T07:11:00.000-08:00

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.

Engaging Students During Problem Solving

2011-02-19T03:31:00.000-08:00

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.

Problem-Solving Approach

2011-01-09T07:06:00.000-08:00

Question
What is a problem-solving approach?

This presentation made in Penang provides answer to this question.

mathz4kidz for parents

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About Mastery of Basic Facts

2011-01-07T07:16:00.000-08:00

Question

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

Answer
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

Question About Use of Notebook

2010-12-29T18:07:00.000-08:00

Hello teacher Ban Har, I am one of the teachers in the course to apply the Singapore method in Santiago de Chile conducted recently. I now have to help 13 teachers next week in using the Singapore method textbooks. I have a question: how to use a note book, apart from the textbooks?

Marcela, a teacher in Chile

Hola Marcela. As we did in the course, we often use one problem in the development part of the lesson. At this point, the students are not likely to be opening the textbooks. They would be given the problem and asked to solve it usually in pairs or groups. They are likely to record their solutions in the notebook. After about 5 to 10 minutes, they will be asked for their responses which are used by the teacher to help students develop the key concept. By the end of the development, if you have attended one of my lessons before, you will see the board filled with different ways to solve the problem. Students can record this in their notebook. Sometimes there is also a conclusion. They will also be asked to write this in the notebook.

At this point the teacher will ask students to do consolidation tasks from the textbooks. These can be done in the notebook.

At the end of the lesson, students can do their reflection (e.g. write a letter to a friend about the lesson) in the notebook.

The notebook is an important tool for students to learn how to organize ideas.

Question About 4 x 4 = ___ + 7

2010-08-17T11:53:00.000-07:00

Second grade students have some difficulties with tasks such as 4 x 4 = ____ + 7. They are students in billingual school and use Spanish and English.

A teacher in Chile

It is expected that students who do not understand the meaning of the equal sign (=) to give 16 as the answer to 4 x 4 = ____ + 7. The teacher can ask students: "What is the value on the right hand side?" (16 + 7) which is 23 and "What is the value on the left hand side?" which is 16. They should realize that these are not equal.

You could read the number sentence this way: 4 x 4 is equal to 16 which is what plus 7.

The task requires a two-step process and students need some metacogntion to complete the task. Use drawing and concrete materials to help the students.

Two Times More Than

2010-08-01T21:05:00.000-07:00

There were some yellow balls and red balls in a box.
The yellow balls were 2 times more than the red balls.
The box had 9 red balls.
How many balls were there in the box?

Should the answer be 36 or 27?

Wendy

This task is poorly phrased. I hope this is not a test item.
Firstly, the quatity to be compared is not clear. Is it the mass, volume, number or some other quantity that is being compared? In this case i believe it is the number. Thus, it should be written as: The number of yellow balls was 2 times as that of red balls. Alternatively, it can be written as : There were twice as many yellow balls as there were red balls.

I believe MOE has addressed this point some years back. At the NIE, this issue should have come up during discussions on test item writing component of the course.

There were some yellow balls and red balls in a box.
There were twice as many yellow balls as there were red balls
There were 9 red balls in the box.
How many balls were there in the box?

In this case, the answer is 27 balls.

We say x is twice as much as y. We say x is 2 more than y. We do not say x is twice more than y. We might say x is 20% more than y.

Questions Posted at Singapore Math in Action Professional Development in Manila

2010-05-20T13:34:00.001-07:00

These are some questions posted at the Singapore Math in Action by Keys Institute for Teaching & Learning.

Q: You said that Singapore Math does not focus on recalling procedures. But isn't math process procedural? Please explain the difference.
A: In the Singapore curriculum, procedural fluecy is important but must be learnt meaningfully. Thus, the long division algorithm is explained using base ten blocks. Or the division of a whole number by a fraction is explained using the grouping meaning of division (how many three-fourths are there in one whole?. Mathematical processes include the use of heuristics and thinking skills. They also include reasoning, communication and seeing connections. None of these are procedural in the sense that the invert-and-multiply strategy is procedural.Mathematics is about sense making rather than recalling procedures.

Q: Many schools in Manila have crowded curriculum. How do you suggest Singapore Math being implemented the way it should be implemented given that kind of national curriculum?
A: Some schools simply use the Singapore curriculum and make sure that by the end of six years their students have mastered all content in the Filipino curriculum. Others use it as a resource to help teachers teach whatever there are in the Filipino curriculum in a meaningful way and to give their students challenging problems to solve.

Q: How do you feel about supplementing the school's curriculum of U.S. Math with Singapore Math? Will it be confusing for the child or would it be helpful for the child to learn different ways of solving problems?
A: Mathematics is the same everywhere. Singapore mathematics textbooks use more visual approaches and can help students learn abstract ideas.

Q: How old are Grade 1 students in Singapore?
A: They turn seven during the school year. They start school in January.

Q: When do you recommend a school to introduce Singapore Math to their students?
A: Any program is best introduce right from the start - Grade 1.

Q: How do you feel about 4th Graders using finger-counting to arrive at addition facts?
A: This should not happen. First and maybe second graders may use finger counting to figure out addition. But these should soon become facts and be remembered. Games such as Salute! seen in the lesson conducted during the seminar should help students gain mastery and fluency in addition facts. See the photograph.

Q: How do you help kids adjust from traditional methods to Singapore Math method especially if they are already in a higher grade?
A: They will need to get use to the use of bar models to solve word problems and they need to get use to more challenging materials. Older students may need some closing of gap.

Singapore Math @ San Diego

2010-05-03T07:51:00.001-07:00

This was the presentation made at Sheraton San Diego Mission Valley Hotel. Yeap Ban Har presented this in the afternoon. William Jackson presented the morning session.

Singapore math @ sheraton

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Presentations at USA in April

2010-04-25T12:02:00.000-07:00

I gave quite a few lectures in Boston and San Diego. The slides are available at www.mathz4kidz.com

On Lesson Study

2010-04-19T11:01:00.001-07:00

I am a teacher in a primary school in Singapore, currently carrying out Lesson Study in our school. I would like to seek your advice on Research Themes.

I gather that to come up with the Research Theme, the team needs to look at the Ideal Profile of Students and the Actual Situation of Students. This, with the School's and Department's Vision and Mision, the team will construct the Research Theme or Research Focus. My question is once this is firmed up, it will form the basis of all future Research Lessons for Lesson Studies, is that right?

Do we need to come up with a different Research Theme for each cycle of Lesson Study or will the Research Theme be the one that all Lesson Studies be based on. For example, my team has discussed and came up with a Research Theme for Mathematics. Will another Research Theme need to be constructed for a different cycle for Mathematics or can we use this Research Theme we have decided on for all Research Lessons for Mathematics.

Ban Har writes: Based on the school's vision, focus of the curriculum and your students' profile, the research theme is constructed. You are on the right track. You will stick with the same reasearch theme for a while until your team feel that there are new areas that need attention.

Teaching Perpendicular Lines

2010-04-03T06:54:00.000-07:00

Question: Should I get students to use set square to draw perpendicular lines in the first lesson or should I give them non-standard tools such as a right-angled triangle?

Hazel, a pre-service teacher in Singapore

This chapter comes after the chapter on angles where students have learnt about right angles. I think before students are taught how to draw perpendicular lines, they need to know what are perpendicular lines so that they are able to check if two lines are perpendicular. I am like to have given them two intersecting lines as well as lines joint at a point and ask them to measure the angles between the lines and to say that those that meet / intersect at right angles are called perpendicular lines. In my opinion, the chapter in the textbook starts too abruptly to get students to draw perpendicular lines without actually being taught what perpendicular lines are.

I like Hazel's idea of introducing other non-standard tools that can be used to draw perpendicular lines. I think a good lesson could involve the students being given some common objects such as ruler, protractor, set squares, triangle tangram set, an index card, ice-cream sticks and asked which of these they can use to draw lines that are perpendicular to each other.

The lesson can conclude with the whole class checking responses that they have generated using the criteria they have learnt.

Implementing Singapore Math & PD

2010-02-14T07:11:00.000-08:00

Can you tell me more about the professional development programm for teachers?

This is an exciting year as far as PD is concerned. You may read about lesson study in Singapore at www.lessonstudyinsingapore.blogspot.com I will also write more on this in a separate entry on a later date.

Suffice to say Ministry of Education in Singapore place a lot of emphasis on professional development. There is a position (called staff developer) in schools where the person (head of department level) is in-charge of professional development of his / her colleagues. This person is one of the head teachers.

What are the things that we must me aware about when we're implementing the program at our schools?

Three things you may want to take note of:

It is best to start from the lower grades. Unless the students have a strong foundation, they may not cope well with the textbooks (which includes challenging problems) if they start in, say, grade five. The teachers also have an easier time adapting to the pedagogy.

Support from the principal and school leaders is very important.

Professional development of teachers must be considered.

Differentiation in Singapore Classrooms

2010-02-14T06:57:00.000-08:00

A friend in another country who requested some information about what happens in Singapore schools. He wanted to know more about the differentiation process. How does this take place in the classrooms? He said that he read that there are lesson after schools for children who have extra needs.

Differentiation is a continuing issue in schools here. chools are always striving to best do this. No wonder there have been quite a few request to conduct in-house training on differentiated instruction.

I think there must be variations on how this is done. But as a system, there are at least two things that happens in every school. One, Learning Support Porgramme (LSM) is a differentiated instruction for primary one and two students who are not entirely ready for the regular instruction. In many schools, these pupils are pulled out from their regular classes and work with a teacher who works with fewer children (not more than 10). The regular class size is 30. The content is the same as the regular programme. However, pupils get more attention because of small class size and the teachers are in a better position to diagnose and remediate. They may also use slightly different pedagogy to enage these pupils - more concrete activities, perhaps.

Another is the Foundation Maths Programme at Primary 5 and Primary 6. This is content differentiation. The students review work from the first four years in an age appropriate way even as they learn a bit of new materials. This is to ensure that the students are ready to join one of the course of study in grade seven (secondary school). They use a different textbook and they are usually in one class. In some schools with very few students in the Foundation Maths programme, they will be pulled out to do maths with another teacher during the mathematics periods.

It is alos not uncommon for teachers to conducrt remedial lessons after 1 p.m. Most of our schools end formal curriculum around 1 p.m. The afternoon is reserved for other non-academic activities, field trips as well as remedial and enrichment lessons.

Singapore Math Training

2010-02-14T06:25:00.000-08:00

I get a lot of questions about the possibility of getting training in Singapore Math. The answer depends on what you are interested to do and which country you are in.

If you are interested in graduate studies, NIE offers masters degree in many specialization, including mathematics education, primary education and secondary education (there are many other fields of specialization). We have foreign stduents who complete the course full-time in a year although a year and a half is not so tight. NIE has both coursework as well as coursework plus thesis option.

If you are interested in Ph.D., you need to be here in the initial part to complete your coursework and to work out your research problem with your supervisor. You may do the research here or in your home country. You will be expected to be back here in Singapore towards the end of the candidature and for your oral examination before the degree is conferred. If you are interested please email me banhar.yeap@nie.edu.sg I can link you up with potential supervisors.

If you are more interested in getting training for classroom teaching rather than graduate degrees these are some options.

Join NIE's post graduate diploma in education (PGDE Primary) where you get pre-service training in two or three subject areas including mathematics teaching. It is one year. If you are not interested in the diploma and just want to focus on mathematics then I think there is an arrangement for you to do this as a non-graduating student.

Join NIE's inservice course. There is a range of subjects available throughout the year. From lesson study to problem solving to action research. A 12-hour course runs over 4 weeks. (once a week).

In some countries, there are institutes already in place. In the US, there is a summer institute every year. Go to http://www.sde.com/conferences/singapore-math/index.asp for details.

Some school districts run theirs. I have been to Scarsdale School District twice to teach a course in their own teacher institute. I believe in the last course there were teachers not from the school district who joined the course.

In places like the Philippines, Indonesia and Chile, the book distributors have been organising training seminars at least once a year.

If you are thinking of running such institutes in your country, I can put you in charge with the book publisher to explore this possibility.

Why Does My Child's Score Drop Suddenly in Primary 5?

2009-08-18T01:46:00.000-07:00

A parent shared her concern with me recently. Her daughter did well in Primary 1 to 4, scoring above 90% in her examinations. She has been coaching her daughter all these years. In her recent SA1 (mid-year examination) in Primary 5, her score dropped "incredibly to just 62%". In a mock test for CA2, she got "just 56%". The parents said she is "scared stiff" and is "perplexed as to how this can happen".

Maybe an explanation of examination structure will shed some light. Up to Primary 4, the proportion of extended problem solving where the tasks tend to be complex, multi-step or novel and students need to show their steps is relatively small. Typically, up to Primary 4, many schools' examination papers allocate 40% of the score for MCQs, another 40% for short-answer questions and 20% for long-answer tasks, the type that tend to be complex, multi-step and sometimes, novel. In upper primary the proportion changes to 20% MCQs, 30% short-answer questions and 50% long-answer questions.

The student concerned is presently scoring about 60%. This means her foundation is strong. This is clear from the fact that in earlier grades her scores had been about 90%. She is capable of problem solving but the type that is now common in upper primary is still a challenge to her. She like many primary five students are still developing this capacity.

My advice is to give her confidence by guiding her through such problems. Ask her scaffold questions so that she can she the intermediate steps that are not obvious to her now. Perhaps in the problems that she uses the model method, the models are more complex now and require strong visualization skills.

Whatever it is do not be overly anxious (I know this is hard for most parents to do) because the anxiety will transfer over to the child and this will not help. Get her to see that her basics are strong. This is probably evident from her test papers. Let her see that the problems she cannot handle are actually rather challenging. Then start her with the simpler ones first. Move to the more challenging ones. Try this and see if you can move her to about 70%. A score of 70% to 80% indicates that a child is within the A grade at the PSLE. Students capable of complex problem solving will move into the 90% range. If I am not wrong the national average for number of students who score A and A* (i.e. above 75%) is about 45%.

Foundation Mathematics

2009-07-23T10:44:00.000-07:00

I really enjoyed your presentations at the conference in Vegas. I'm curious about one of the books you mentioned and perhaps where I could find it. You said there is a book for the Foundations Mathematics Program.

Karen, 5th grade math teacher in Austin, Texas

By the end of Primary 4, students who have not acquired adequate basic foundational knowledge are recommended to do an alternate program. This is referred to as Foundation Mathematics, offered to Grades 5 and 6 students. Less than 10% of Singapore students do this program. This program help students review a lot of Grades 1 to 4 materials in an age-appropriate manner with some Grades 5 and 6 topics included. This program is suitable for US Grades 5 and 6 students who have not had Singapore Math background. The books are available at http://www.singaporemath.com/Math_Works_Coursebook_5A_p/mwc5a.htm. There are two textbooks (5A and 5B for Grade 5) and two workbooks per grade level.

Language Issue

2009-07-18T23:24:00.000-07:00

When we teach mathematics, we introduce two foreign languages to our learners (pretty hard for them) - the language of mathematics and the English language, because here in the Philippines, English is our medium of instruction. Other issues came out, that learners would learn best if they would use their mother language. I think this is one of the major issues now, not only here in our country but also globally. An example is Malaysia.

Jeniffer in the Philippines

There is no guarantee that using one's mother tongue results in high achievement in mathematics. Japan and Korea use Japanese and Korean to teach mathematics and their achievement is high (e.g. in TIMSS). Thailand and, until early 2000s, Malaysia use Thai and Malay to teach mathematics and their achievement is not high. Singapore use English, not the students' mother tongue and the achievement is high. There are other more important factors than the medium of instruction. In my opinion, if the medium of instruction is used right from Grade 1, students will just pick it up. Students from non-English speaking homes may need some additional help. In Singapore, we have Learning Support Programme in Grades 1 and 2. I feel that Malaysia should not say that their attempt in using English to teach mathematics has failed. If their politicians have read Michael Fullan's Six Secrets of Change, they will understand that when a change is implemented there will be a dip in performance before it increases again. I feel that they should have given the change a longer time to happen before deciding to abandon it. Incidentally, I had the good opportunity to listen to Michael Fullan this morning at a conference in Las Vegas. He was giving lectures to US principals who are attending the differentiated instruction conference.

Two Questions

2009-07-18T22:56:00.000-07:00

May I know your stand on these issues, "Stop Teaching by Telling'' and "The Principle of Equity (in mathematics classes)."

Jenny, a teacher in the Philippines

Often teachers teach by one of these methods: telling, coaching, modelling and providing. There is probably a place for each technique. Generally, I would urged teachers to help students develop conceptual meaning of procedures they learn. For example, to lean 3 divided by 1/2, I would suggest teachers use pattern blocks or pictorial representations and ask students questions such as "How many halves are there in 1 whole? So, how many halves are there in 3 wholes?" From there students can see that the answers are 2 and 2 x 3, respectively. They understand why 3 divided 1/2 is the same as 3 x 2.

By using differentiated instruction, teachers are able to provide every child with equal opportunity to learn. For example, in practising multiplication, teachers may allow struggling students to use concrete materials to derive multiplication facts while challenge able students to spot patterns in the multiplication tasks or to use basic multiplication facts 7 x 7 and 7 x 5 to do say 7 x12.

Concrete Representations in Mathematics

2009-07-18T22:45:00.000-07:00

My co-teacher asks me to e-mail you because she attended a conference in Manila, the Philippines in May 2009. What are the different processes in teaching concrete ideas in mathematics? Thank you so much.

Maria

I assume you are asking about the use of concrete representations to help students learn abstract mathematical ideas. This is the CPA Approach, as we call it in Singapore.

It is important to match the cognitive processes required in handling the abstract idea with the cognitive processes used in handling the concrete materials. For example, in doing addition within 20 (say, 7 + 5), one abstract ideas we want students to have is 'making 10'. Students should be able to visualize that 7 + 5 is the same as 10 + 2 (making 10 from 7 by moving 3 from 5). A suitable concrete activity is to use tens frame. Use two tens frames - one with 7 counters and another with 5 counters. Then get students to move 3 counters from the second frame to the first. The cognitive processes match.

The photograph shows tens frame made by public school teachers involved in the LEAP Project in the Philippines (December 2008, Ateneo de Manila University).

Model Method

2009-07-01T07:54:00.001-07:00

I'd like to ask about this problem:

Ahmad and Mei Ling saved $800 altogether.
A quarter of Ahmad's savings was $65 more than a fifth of Mei Ling's savings.
How much more money than Mei Ling did Ahmad save?

Please explain the answer using model

Tendo, a teacher in Indonesia

For some reason the photograph does not show up.

Draw a unit bar for a fifth of Meiling's savings. So, a quarter of Ahmad's saving is this unit bar plus a bar that stands for $65. Now draw the whole amount of Ahmad's savings (four units plus four $65) and the whole amount of Meiling's savings (five units). These add up to $800, right? That means nine units plus $260 is equal to $800. Can you finish it up?

A Multiple Problem

2009-06-18T00:31:00.000-07:00

I am a two-digit number. When I am divided by 8, there is a remainder of 3. When I am divided by 9, there is a remainder of 4. I am less than 80. What number am I?

Jane, Indonesia

One way is to guess and check. Make an intelligent guess and check if both conditions are met. I can guess 27. 27 divided by 8 gives 3 remainder 3. 27 divided by 9 gives no remainder. So is the number 27?

I think it takes time to guess this way. Let's use logical reasoning. The number is 3 more than a multiple 8 and 4 more than a multiple of 9.

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80.

Three more than a multiple of 8: 11, 19, 27, 35, 43, 59, 67, 75 (no need to try 83 and beyond)

Multiple of 9: 9, 18, 27, 36, 45, 54, 63, 72

Four more than multiple of 9: 13, 22, 31, 40, 49, 58, 67, 76

You got the solution, right?

How about a different way? If you know some algebra, the number is 8m+3 or 9n+4 where m and n are whole numbers. 8m+3 = 9n+4 or 8m = 9n + 1 which gives a possible solution of m = 8 and n = 7. Thus, the number which is 8m+3 can be easily found.

What if the last condition that it is less than 80 is not given?

Can you make up a similar interesting problem for the others to solve?

Pre-School Numeracy & Assessment

2009-06-04T02:14:00.000-07:00

We are writing you from Santiago, Chile. We are using Earlybirds Kindergarten Mathematics. The doubt is that in Book A, to evaluate classification, is it necessary to evaluate all the previous steps such as "different things", "things that are used together", "things that do not belong" etc.? Or can we evaluate only the final concept, that is sorting by the three attributes.

We have to tell you that this is our first year using your method in Mathematics, and, at the beginning , we had some problems, because of the language (our students don´t speak English at home, they learn it only at school). But now, we have used the book for 4 months, and we think the students, and we, the teachers, have learned a lot from it.

Anamaría, Chile

In Kindergarten, we do not want to overwhelm children with assessment and evaluation. In this unit students have learnt how to match things according to some attribute (being able to say two animals are the same despite differences in size, color, orientation; being able to say two things are the same even though they are drawn differently; matching by colors; matching by patterns). Some where in between they apply this to pick the odd item out among, say, four items. Later, they learn to classify according to more abstract attributes such as function (what they are used for). Again, the apply these skills to pick the odd one out. Finally, the use the skills to classify items into two groups according to given criteria and their own criteria. In the evaluation, we assess and evaluate if the children are able to apply these skills in picking the odd one out and in classification.

There is no need to evaluate every sub-skill. However, if any child cannot complete the main task (odd one out and classification) then teachers may want top check if they can do the sub-skills.

Test Items from Singapore National Tests

2009-06-03T01:09:00.000-07:00

I understand that somewhere in your website I can access past primary-level mathematics tests, but I cannot find them. Do they exist, and can I get access to them? I am learning about Singapore Math.

Lee, a math coach in Utah

These are not available online. Released items from the tests are compiled into a book every year. For example, the most recent book consists of released items from 2004 to 2008 tests. About three-fourths of all items are released each year. This book is not available outside Singapore. I believe this is an agreement between the copyright holder of the test items and the publishers. In Singapore, they are easily available in any Popular Bookshops. If you are outside Singapore, get a friend to help you buy a copy.

The national test at the end of primary schooling is the Primary School Leaving Examination. Students are tested on English, Mathematics, Mother Tongue Language and Science. For mathematics. most students do the Mathematics test. A small proportion do the Foundation Mathematics test which focuses more on basics and less on problem solving.

Students complete 15 multiple-choice items (20 points), 20 short-answer items (30 points) and 13 long-answer items (50 points) which includes some challenging tasks. The first paper (50 min) includes the multiple-choice items and 15 short-answer ones. The second paper (1 h 40 min) includes the other items. Students can use a calculator in the second paper. This format will be used for the first time in 2009. Previously, these items were in one 2 h 15 min paper and students do not use calculators (and the numbers are not tedious to compute).

Word Problem

2009-05-25T16:38:00.000-07:00

I'm the mathematics coordinator of 3rd and 4th grades of my school in Chile. We are using the My Pals are Here Series. But I have a doubt, concerned to the fact that even though we are focused on solving word problems using the models methology and it makes sense to the girls when we are working together, they are still having problems when they are working alone, specially during tests. They start right away with operations but most of the time it is wrong, because they didn't visualize the entire problem. Do you have any suggestion? Should we continue with next chapter or work more on problem solving with additions and subtractions?

Question posted by Paula, a mathematics co-ordinator in Chile

As you have mentioned, when the students start straight away with the operations they are often wrong. They need to comprehend the problems well. Drawing a model will help them understand how the information are related. In simple one-step problem, it may not necessary to do so. But in a problem with a lot of information, this becomes essential for average students. Otherwise although they can read the word, they do not comprehend the information.

Also in multi-step problems, the students may not have the ability to monitor their thinking. This is metacognition. When we teach word problems, we should model and coach rather than explain. That way, we help them in developing the ability to think through the many steps in a problem.

Speed Problem...Again

2009-05-11T03:21:00.000-07:00

A car needs 7 hours to travel from Town X to Town Y. A motorcycle needs 8 hours to travel from Town Y to Town X. The car leaves Town X for Town Y and the motorcycles leaves from Town Y to Town X at the same time. How long will it take for the car and the motorcycle to meet?

Angie

Speed Problems are frequently brought up. There are earlier entries discussing Speed Problems. See below.

So, how long will it take for the car and the motorcyle to meet. The standard joke is that we hope they don't!

That aside, we need to assume that the speed of the two vehicles are constant. If that is so then in an hour, the car travels 1/7 the distance in an hour and the motorcycle travels 1/8 the distance in an hour. The problem is solved when the distance travelled by the car and motorcyle add up to 1 whole. In an hour, total distance covered by both is (1/7 + 1/8) of XY. This works out to 15/56 of XY. In 2 hours, it is (2/7 + 2/8) of XY or 30/56 of XY. In 3 hours, 45/56. In 4 hours, 60/56. They would have passed each other in 4 hours. Can I leave it to you to complete the last step of the solution. It is by no means trivial but there are enough leads already.

Request for Presentation Slides

2009-05-08T04:04:00.001-07:00

I am a 5th grade teacher in Fayetteville, NC. I had the amazing opportunity to attend your session at the NCTM Conference in Washington, DC a few weeks ago and was truly inspired! If possible, would you be able to email me a copy of your handouts and powerpoints used in your session? I would greatly appreciate it! Thank you for your time and amazing inspiration!

Laura, an American teacher

The presentation slides at the NCTM Annual Meeting & Exposition are available at http://math.nie.edu.sg/t3/downloads-conference.htm Look for the conference that you are interested in and click on the pdf. The slides should download. The slides for my other presentations are also available here.

A Problem from a Hong Kong School

2009-04-27T08:04:00.000-07:00

A mom shared that her P1 daughter in Hong Kong was posed this mathematics problem:

AB + B = BA

A = ________B = ________

How do you teach a P1 child algebra?

Lily

In formal algebra, ab means the product of a and b. I do not think that is what the problem is about. In this problem, I believe A and B represent digits and AB is a two-digit number which when added to B gives a two-digit number which has the tens and ones digit of the original number (AB) reversed. In this case the letters are not used in the same convention as formal algebra. I would recommend that this problem is presented orally to grade one children. The teacher might say, "Each letter (or shape) stands for a digit. The same letter stands for the same digit. Different letters stand for different digits. I have a number, the tens digit is A and the ones digit is B (Teacher writes down AB). When it is added to a number B (Teacher writes down AB + B), the total is a number with B as its tens digit and A as its ones digit. (Teacher writes down AB + B = BA) Find the digits A and B."

I would illusrate with an example say 12. What is A in this illustration? What is B? A is 1 and B is 2. So, in this problem AB + B (which is 12 + 2) is supposed to be BA (21). But it is not, right? So AB is not 12. I suppose guess and check is the best strategy for grade one children to use.

Advanced students or older students may reason this way: Is A an odd or even digit? Yes, it must be an even digit. Why? Did you notice that B + B = A. Sure it could be 1A as well. But not 2A or 3A or 4A and so on, right? Since the final sum is BA, AB + B must involve renaming. Why? Otherwise the tens digit in the sum is the A, isn't it? So B must be 6, 7, 8 or 9. And A is one less than B. Think about this one! Hence, A is 5 when B is 6, A is 7 when B is 8 - both not possible. Why?

Hence, A = 6 and B = 7 or A = 8 and B = 9. Checking 67 + 7 = 74 and 89 + 9 = 98. I think the solution is A = 8 and B = 9.

Secondary students may solve it algebraically: 10A + B + B = 10B + A or 9A = 8B or the ratio of A : B = 8 : 9. For digits, A has to be 8 and B has to be 9.

I don't think P1 children are expected to do the algebraic solution or even the reasoning based on number properties. They are most likely able to solve it by guess-and-check.

Another Speed Problem

2009-04-13T04:53:00.000-07:00

I have a word problem about speed that I cannot explain to my pupils easily.

Car A and Car B left Town X for Town Y at the same time. Car A was travelling at an average speed of 80 km/h and Car B was travelling at an average speed of 60 km/h. Car A was leading Car B by 8 km for every 1/6 of the distance from Town X to Town Y. Find the distance between Town X and Town Y. One of my friends solve it like this: 6 units x 8 km = 48 km and 80 km - 60 km = 20 km. 80 : 20 = 4. Hence, the distance from Town X to Town Y = 4 x 48 = 192 km. However, the solution is too difficult for my pupils.

Charmaine's Suggestion:

Car A travels 20km more than Car B in an hour. (80-60)

Since Car A leads Car B by 8km for 1/6 of the journey, it leads by a total of 8*6 =48km for the entire journey (6/6).

Thus, time taken for the whole journey by A: 48/20hours = 12/5 hours (leave in simplest improper for easier calculation later...)

Distance between X and Y is thus 12/5*80 = 192km.

Seow's Suggestion

http://farm4.static.flickr.com/3648/3457754184_35d7fcdc47_o.jpg

Speed Problem

2009-04-11T06:00:00.000-07:00

I teach maths in Grade 6. There is a word problem in the maths book that I cannot teach using the model method that is usually more suitable for my pupils. The problem: A car and a lorry were travelling towards Town A. The car overtook the lorry when they were 90 km away from Town A. The car arrived at Town A 1/2 h earlier than the lorry while the lorry was 30 km away. Find the average speed of the lorry. Find the average speed of the car.

Gita

Please do not think that the model method can be used for all problems. That is not the idea. We want students to learn a variety of heuristics and they should apply it accordingly. Speed problems are rarely solved using the model method. A line diagram is more useful. Please see any Singapore Grade 6 books for such line diagrams.

In the problem posed, the lorry took 1/2 h to finish the last 30 km. So the average speed of the lorry is easily found (60 km/h). With 90 km to go, both the car and the lorry has travelled the same distance from the spot where the lorry started - that was when the car overtook the lorry. The car must be faster but started later.

I want to suspend the solution for a while. I invite readers to continue to solve the problem. Post further question if necessary.

2009-04-10T04:32:00.000-07:00

I have a word problem: The price of a pen was $5, The price of a pencil was $2. Miss Lee bought a number of pens and pencils for $26. How many pens and pencils did she buy?

The day before yesterday I taught my pupils this: I made a list for the pens : 1 x $5 = $5 , 2 x $5 = $10 and so on. And also a list for the pencils : 1 x $2 = $2 and so on. Miss Lee bought 4 pens and 3 pencils because (4 x $5) + (3 x $2) = $26. However, I realize that this method cannot be used in big numbers.

Merry

The method you used is called make a list. It is a common problem-solving heuristic. Please continue to use it with your younger students. I wonder if Miss Lee could also buy 2 pens and some pencils. I know 5 pens is not possible because the money left is an odd number $1 and the price of a pencil is $2. Similarly, 1 pen or 3 pens are not possible. Students learn reasoning.

If students know algebra, they can set up equation 5x + 2y =26 where x is the number of pens and y is the number of pencils. As there is only one condition, you still need to use guess-and-check to solve this equation. Unless the problem says something about x + y.

If the value is not 26 but larger then the equation is 5x + 2y = k where k is the large number. A graph can be plotted and possible solutions seen on the graph. (With larger k the number of solutions increases).

For the young students, the method you use is probably the best. When they are older, they will learn to solve the same problem with larger k values.

Goggles Problem

2009-04-09T03:02:00.000-07:00

This problem is from Fairfield Methodist Primary School, used in a seminar for their parents.

88 children took part in a swimming competition. 1/3 of the boys and 3/7 of the girls wore swimming goggles. Altogether 34 children wore swimming goggles. How many girls wore swimming goggles on that day?

I have read a word problem you wrote in the blog that asks for the number of girls who wore goggles. It is rather difficult for me. I am in Grade 3. However, I just tried to solve it and I found the answer is 21 girls wore goggles. Is it right? In solving it, I was helped by one of my cousins who is in Grade 7.

Jane, a third-grade student in Indonesia

Please go to http://math.nie.edu.sg/T3/downloads/2009%20Parents%20PSLE.pdf to view a powerpoint presentation of the model used. Your answer is correct! Keep it up.

Classroom Management

2009-04-05T14:50:00.000-07:00

I've been using your method so far and it is amazing how kids learn, but there is a problem in the Chilean concept of discipline. Chilean kids tend to be very immature.

Carola, a teacher in Chile

It is often said that classrrom management problem will disappear when kids are engaged in the learning process. Hopefully with time, when the kids are engaged they will be better in the class, in terms of discipline. But my personal experience with teaching four research lessons in Chile was very good. Besides my observation that some kids in the public schols do not develop basic skills such as addition well enough, they are otherwise fantastic - just like kids in Singapore and everywhere.

In Singapore schools, discipline is emphasized. We still have difficult cases with a small number of students.

The photograph show a demonstration lesson done in Chile for a seminar organized by Ministry of Education Chile. The fourth graders were doing an exercise on pictorial and symbolic representations of fractions. Despite the language barrier, I did not find the kids difficult to manage. The Minister of Education opened the seminar.

A Nine-Year Old Likes Model Method

2009-04-05T03:57:00.000-07:00

I am 9 years old and I live and go to school in Jakarta. I like solving maths word problem by the model method. I go to school that teach maths in the Indonesian language. My mother bought me some Singapore maths book two months ago. I like the model methods that are given in the books. My teacher has never taught maths using the model method. My English is not so good so I must look up a dictionary for difficult words. Can I ask you for help if I find some word ptoblems difficult?

Jane, a nine-year old in Jakarta

If Jane pose some questions that she finds difficult, I will share it here. But here is one that comes from a school in Singapore. Try it!

Word Problem on Speed

2009-04-03T03:14:00.000-07:00

I have a word problem that I cannot solve. The problem is: Carson and Daniel started driving from the same place but in opposite directions. After 5 hours, they were 60 km apart. Carson's average speed was 12 km/h less than Daniel's. What was Daniel's average speed ?

A teacher in Indonesia

Every hour, who travels a longer distance? Daniel, isn't it? How much further? That is correct - 12 km every hour. That means after 5 hours, Daniel has travelled 60 km more than Carson. Since they are 60 km apart after 5 hours - what does that mean? It seems that Carson was not moving at all and Daniel was travelling at an average speed of 12 km/h - which seems a little slow!

Is there a flaw in the reasoning? Or the problem was not well-posed. Comments?

Freudenthal Institute

2009-04-01T07:18:00.000-07:00

I got this message from Freudenthal Institute. You may be interested if you are a school teacher in science and mathematics.

Hereby we would like to let you know that the Utrecht Summer School in Science and Mathematics Education at Utrecht University will be held from August 17 - 21, in the Netherlands. Attached you will find an invitation to participate. Target group: Science and mathematics teachers with an MSc and MA degree in one or more of the Sciences & Mathematics as well as proficiency in English.

Looking forward to meeting you in the Netherlands,

With kind regards,

Jaap den Hertog
Freudenthal Institute for Mathematics and Science Education
Utrecht University
The Netherlands

http://www.utrechtsummerschool.nl/
http://www.science.uu.nl/summerschools
http://www.fi.uu.nl/fisme/en/
http://www.science.uu.nl/summerschools/appliedsciences/

USSE 2009
c/o Ank van der Heiden
Freudenthal Institute,
PO Box 9432 3506 GK Utrecht The Netherlands
T: +31 30 - 263 55 55 F: +31 30 - 266 04 30
Email: A.vanderHeiden@fi.uu.nl

Deadline for Application: 1 May 2009

Base Ten Blocks

2009-03-23T21:43:00.001-07:00

I am preparing materials for teaching Grade 3 math using My Pals are Here 3A. I can use paper cut-out versions of the base ten blocks for tens and ones and possibly hundreds but I am having a problem with the thousands place. For Grade 3, the numbers go up to 10 000. Since we do not have the physical base ten blocks available, I am considering other options to represent the thousands, hundreds, tens and ones. I was considering making sketches of marbles in a jar. It is easy to make sketches of bottles of ten marbles, hundred marbles and thousand marbles. I am worried about the fact that the model may not be a proportionate one and will affect students' understanding.

Geoff, a teacher in the Philippines

Since you are using sketches, why not use diagrams of thousands, hundreds, tens and one (as in the textbook). I suppose the third graders can do without the actual base ten blocks as they have used it in first and second grades. The idea of converting a ten into ten ones etc should be well developed by the third grade. However, a pictorial representation is still necessary to help students to visualize numbers and operations such as addition, subtraction, multiplication and division which they will learn later in the third grade.

I suggest you substitute the actual base ten blocks with pictorial representation such as the ones you see in the textbooks.

Fraction of a Fraction Word Problems

2009-03-17T19:48:00.000-07:00

One of my student teacher doing his practicum has found that teaching topics that has been dealt with by his classmates during micro-teaching to be easier. This is a good reminder to me as a teacher educator of the importance of micro-teaching in a teacher preparatory programme.

Somehow, students find difficulties in solving the word problems related to the topic of fraction of a fraction. To be more specific, they have problems with tasks such: Ben sold 7/12 of his poultry. Of his unsold poultry, 3/5 were chickens and the rest were ducks. What fraction of the all the poultry were the unsold ducks?

Students have difficulties with the word remaining. Although most of them can solve it easily using the model method, some cannot understand the remaining part when asked to solve it just by using a number sentence.

Posed by Khai

This situation is a reminder why it is important to use pictorial representation (such as the model method) before moving on to symblic ones (number sentence method). The model method is a good link to the abstract number sentence. I would advise teachers to continue to use diagrams to model the situations and leave it to students to make the leap to the abstract symbolic representation. The situation posed is a good starting point. For advanced students, the 3/5 can be changed to say 7/10.

Research Network

2009-03-13T07:08:00.000-07:00

What are some avenues to keep me connected to mathematics education?

There are many avenues. One of the is activites of ICMI - the international group that focuses on mathematics instruction / teaching. One of their major activites is the once-every-four years ICME. The last one ICME-11 was in Mexico. The next one ICME-12 is in Korea (2012). The proceedings of ICME-10 in Copenhagen, Denmark is available. See Niss. M. & Emborg E. (Eds.). (2008). Proceedings of the 10th International Congress on Mathematical Education, Roskilde, Denmark: IMFUFA. An electronic version is downloadable from: http://www.icme10.dk/proceedings/pages/side01main.htm

The ICME-12 congress will be held in Seoul, Korea, on July 8-15, 2012.

The printed version (together with a CD-ROM with all the regular lectures) costs 50 euros (including shipment). The order form is available at the same address.

Personally, I have been to every ICME since Japan (2000). This conference helps one updated about the latest developments in all areas of mathematics education research. It also offers a chance to catch up with overseas collaborators and form new networks. It is a good sounding board for PhD students to get other opinions about their research problem. I have always encouraged my own to do so.

Division of Fractions

2009-03-13T02:30:00.000-07:00

I was one of the participants in the training at Sampoerna Foundation. I'd like to ask about fractions 2/3 : 5/8 = 2/3 x 8/5. How can I explain to the students the reason why the division symbol becomes multiplication and why the second fraction is inverted.

Tendo, a teacher in Indonesia

There are several variations to this task.

1. 2/3 : 1/3

2. 2/3 : 1/6

3. 2/3 : 1/8

4. 2/3 : 5/8

Students have learnt two meanings of divisions in whole numbers. For example 12 : 4 has been modeled by 12 things shared among 4 persons as well as 12 things put into groups of 4. For the latter, we ask "How many 4s are there in 12?"

We can use this meaning of division to teach 2/3 : 1/3. How many 1/3s are there is 2/3? This is obvious. The answer is 2. For 2/3 : 1/6, students need to be able to see that 2/3 = 4/6. Then the answer to how many 1/6s are there in 2/3 becomes obvious.

For 2/3 : 1/8, students know that 2/3 = 16/24 and 1/8 = 3/24.

It is easier to do 2/3 : 1/6 which is the same as 16/24 : 4/24. There are four 4/24s in 16/24.

In 16/24, there are five 3/24s (which is equal to 15/24) and another 1/3 of it. Thus, 2/3 : 1/8 = 5 and a third.

Can you explain 2/3 : 5/8?

Research

2009-03-06T18:31:00.000-08:00

I am doing my thesis and I am looking for some related studies on the levels of conceptual understanding and levels of thinking skills in mathematics.

Arnold, a teacher in the Philippines

I understand you are interested to measure students' levels of understanding in mathematics. There are several frameworks that you may be interested in. Understanding by Design (Wiggins & McTighe) talks about different facets of understanding. There is a Wikipedia entry on this. Richard Skemp (http://www.skemp.org.uk/) has written about instrumental understanding and relational understanding. This is another way to look at different types of understanding that students have in mathematics. These two will be good starting points for you to consider how to investigate students' level of mathematical understanding.

As for levels of thinking, perhaps frameworks that discusses levels of cognition are useful. The good ol' Bloom's taxonomy is one possibility - it has since been updated. Otherwise, you may be interested to read about SOLO taxonomy (Biggs & Collis) - there is a Wikipedia entry on this too. Otherwise see

http://www.learningandteaching.info/learning/solo.htm.

Marzano's dimensions of thinking may be of interest to you.

Multiplication & Division Algorithms

2009-03-05T02:04:00.000-08:00

I have a student who is very good in mathematics. But he does not follow the steps like regrouping to divide and multiply. He writes down his answer directly. My question is must he follow the steps? Must he show the working in vertical format?

Tia, a teacher from Indonesia

He should not need to. After all we want them to use mental strategies for computations. He has been able to do that. The use of paper and pencil should be a temporary help for children who cannot do it using mental strategies (because their number sense and metacognition are not yet well developed). For example to divide 96 by 8, we want students to see that 96 is 80 and 16. Both 80 and 16 can be divided by 8 mentally to give 10 and 2. hence, 96 divided by 8 is equal to 12. A 12-year-old should not need to use paper and pencil to do a computation such as this. Similarly, to multiplyu 39 and 6, we want children to use the product of 40 and 6 to obtain the product of 39 and 6. The ability to do so tells us that the child is able to make connections between the two and has a strong number sense.

Learning Support

2009-03-03T06:47:00.000-08:00

What are the support given to children who have difficulties with mathematics in Singapore schools? In my school in Indonesia we have a learning support unit (LSU). In my class, I have 3 children that have learning disorder (LD), and also children that have problems with concentration. I already done some reteaching. Once in a week they will be learn with the LSU team. But I feel that these measures are not entirely effective. Please give further suggestions.

Sinta, a teacher in Indonesia

In Singapore, some teachers have received in-service training to handle special needs children. However, not every teacher has received this training. In grades one and two studnets who are struggling with mathematics can join a learning support in mathematics programme (LSM). In LSM classes, children received more attention from the teacher because the class size is small, with fewer than ten children in a class. They often cover the same materials as children in regular classes. In some cases when the other children learn with their regular teacher, LSM children go to another room to learn with the LSM teacher. In other cases, the classes are in addition to regular classes. For other children who may have difficulties with certain topics, the teacher may spend extra half an hour or an hour a week to have remeidal lessons with them. In some schools, the remedial is ad-hoc. In others, it is planned and the children have remedial on a particular day of the week.

The important thing about remediation is diagnosis. Teachers must know exactly what difficulties a child has before we can do effective remediation. So, not just remediation. It is always diagnosis and remediation. For example to diagnose what difficulty a child who cannot solve word problems has, we may use the Newman procedure. Please read about Newman procedure in another blog entry.

Negative Numbers

2009-03-03T06:43:00.001-08:00

I have difficulties explaining negative numbers to my primary-level students. Thermometer scale is often used. Negative numbers is part of the Indonesian national curriculum.

Sinta, a teacher in Indonesia

I know that in some countries' curriculum, negative numbers are introduced in primary levels. In Singapore, we only introduced it at secondary level (grade seven). The number line strategy is suitable because it is visual. So thermometer or water level in a dam are suitable. With water dam analogy -2 can be described as 2 m below the land level. In that way we can explain why -2 + 1 = -1.

Number Patterns

2009-02-27T11:21:00.000-08:00

I have a problem taken from a primary five worksheet which I have been unable to solve. I need your advice: List three terms to continue a pattern in each of the following. 5, 6, 14, 32, 64, 115, 191, ...

Linda (Singapore), a parent of a primary five boy

Why don't you ask your child to try and listen to his reasons. There are different possible answers depend on the relationship that we give to the numbers. This is the case when the blanks are at the end. Although it is good discussion problem, as an examination item it needs to be less ambiguous.

My son managed to solve the problem for me.

List three terms to continue a pattern in each of the following. 5, 6, 14, 32, 64, 115, 191, 299, __, __

This is his explanation but I am not sure of it is correct. Please advise.

Find the difference between each of the above numbers:

1 8 18 32 51 76 108

Find the difference between these numbers again:

7 10 14 19 25 32

Find their difference again to get a common difference:

3 4 5 6 7

and continue again to get

1 1 1 1

So to find the next three terms, just add the numbers from 1 , e.g. 1 + 7 + 32 + 108 = 299 and then continue to find the next difference again.

Linda (Singapore), a parent of a primary five boy

You should congratulate your child. He has done a good job.

Research

2009-02-23T12:37:00.000-08:00

My research area for my graduate studies is in educational psychology. Could you suggest what particular variables I can investigate on? After some experience with Singapore mathematics, I am exploring to investigate its effectiveness in learning mathematics.

Rochelle, a teacher in the Philippines

If you are considering experimental studies then you might investigate the effects of the concrete/pictorial/abstract approach on mathemnatics learning which could include achievement in procedural tasks, familiar problem-solving tasks and novel problem-solving tasks. You may choose a unit of study for the experiment. You can compare the results to (a) a comparable control group being taught in the so-called traditional way and (b) the same experimental group being taught a different unit in the traditional way (before the experiment). If you are considering case study then you might want to investigate cognitive, metacognitive and affective behavious of students in different grade levels solving problems using the model method. By varying the tasks systematically, you will also be able to investigate task effects. For the first suggestion, the theoretical underpinnings can be found in Piaget and Bruner's work. For the second suggestion, I will see if I can somehow post some studies conducted in Singapore on the model method. You might find useful articles in the journal The Mathematics Educator. See math.nie.edu.sg/ame

Division of Decimals & Mixed Numbers

2009-02-12T05:28:00.000-08:00

I would like to ask why is it that Singapore primary curriculum does not include division of decimal by a decimal and division with a mixed number divisor or dividend for grades five and six respectively? We have been teaching these to our fifth and sixth graders in the Philippines.
Virgie, a teacher in Manila

The Singapore curriculum attempts to focus on as few ideas as possible so that students develop a strong foundation for mathematics in grade seven and beyond (secondary school). Topics that are not essential will be taught at later grades. For example, while 0.6/3 can be taught as sharing (0.6 is shared equally among three persons), 0.6/0.2 needs to be taught as grouping (How many 0.2s are there in 0.6?). Conceptually, primary students have done both for whole numbers (grade one) and for fractions (2/3 dividied by 4 in grade 5, 2/3 divided by 1/6 in grade six). In the Singapore curriulum, 0.6/3 is taught in grade four. In principle, 0.6/0.2 can be taught in grade six. However, in order not to burden the students with too many things, why not do it a year later? In fact, we are hoping that students are able to extend what they learn in 2/3 divided by 4 to 0.6 divided by 0.2. This is the spirit of Teach Less, Learn More which the Singapore Ministry of Education encourages schools to do in implementing the curriculum. The same explanation goes for division involving mixed numbers.

Number Bonds

2009-02-07T03:32:00.000-08:00

My students seem to have difficulties with number bonds. Can you suggest some ways to make it easier for them? Also, is it necessary for students to write the steps when they do number bonds they way the textbook (see photo) represent it?

We teach number bonds as a preparation for students to learn their addition facts. Addition facts are addition involving single-digit numbers such as 4 + 6 = 10. The suggested way to develop number bond is to use concrete materials such as unifix cubes like the ones shown on the textbook in the photo. Ask the children to show 4 cubes and 6 cubes and ask them to tell the number of cubes. Initially, they may need to rely on counting to respond correctly. After a while, the results will be remembered. Other additional support to help them remember is to review them for a few minutes each day. Or put up poster of common numbers bonds in the classroom for the childrent to see them. Or you can write songs about 10 is 3 and 7 for them to sing!

As number bonds is to help children with mental computations, the written steps are not necessary for students to write. In fact, sometimes I find the kids get confused by it. The representation is more for teachers to see the way to split the number up. In teaching children to add 5 + 7 + 6 as shown in the textbbok, with the help of unifix cubes, show children how you would break 7 into two numbers so that they can make 10 with 5. Say, "I want to make 10. So I break 7 into 5 and 2. This 5 and that 5 make 10." Then, continue with simple addition, "This 2 and that 6 make 8." Finally, "What does this 10 and that 8 make? 18? That's great!" Let them try the same process with breaking 6 to make ten with 7. You must give children concrete materials to model the process. Coins, seed and other common materials can be used if you do not have unifix cubes. Remember that this is done in the second half of the year. In the first half, they have learn how to do the first step - breaking numbers up in different ways and making ten.

Question posed by Yunia (Indonesia), a grade one teacher.