Friday, September 4, 2009

AKM101 Lesson 4

We had four lessons and for the next few weeks we will be having online lessons. Today, we re-examine the issues of teaching mathemeatics versus teaching children. I am glad to hear many views including if the national examination assess good habits of mind and spirit of innovation and enterprise. It is popular rhetoric that all the things outlines in various initiatives are fine and good but the examinations do not value them and hence teachers do not place emphasis on them. That is what this course is about - for the participants to re-examine this beyond the popular rhetoric, to reconsider one's present point of view and points of views of the other participants and to examine the issues more deeply. Thus two questions - one, how to help students reach a certain level of achievement that will allow them to progress towards becoming a productive citizen of the country and the world and, two, as teachers, should we go all out to ensure highest grade possible, even at the expense of teaching the child values and developing his or her confidence to face the world or moulding their personality as a well-adjusted youngsters?

Friday, August 14, 2009

AKM101 Lesson 1

We dealt with recent initiatives in primary mathematics. Today I used two case studies to get students to think how to bring to life words in initiative documents such as the curriculum document. The first case study was a problem that is different yet connect to a pattern that is taught in the curriculum. In the curriculum, children deal with patterns such as ABCDABCDABCD... type patterns where they are expected to be able to find the n th letter in the pattern. In this lesson, we did a very different pattern. Students were asked to propose solution methods and evaluate if the methods are generalizable or not. The second case study was on teaching area through problem solving, using a method that is commen in the textbooks. The students looked at the curriculum document and were asked to think about the rationale for teaching mathematics. Students were asked to make tetrominoes and compare them. Along the way, the basic concept of area and deeper concepts of area emerged. I was excited because the students were giving their views and opinions and asking questions on issues we discussed. I am looking forward to the next lesson. The students were asked to think about what is mathematics teaching all about according to the official documents and according to their own views - what it is and what it should be.

DCM200 Lesson 1

Teaching of Area. I tried to show how teaching through problem solving can be done for students in Primary 3. We did Tetrominoes (Make different shapes using exactly four square tiles.), Quadrilaterals (Make different quadrilaterals using exactly three square tiles and two half-square tiles), Area is Five Problems (Use the geoboard to make quadrilaterals of area 5 square units.). We ended by exploring the Pick's Theorem using the findings of the last problem. I wished I had more time for the students to explore the relationship between area and number of 'dots'.

Sunday, July 12, 2009

Las Vegas Lessons

Latest: The slides for the keynote lecture for National Conference for Singapore Math Strategies are already available at Other slides will be available at soon.

Riviera Hotel and Casino will be my classroom for four days. I will be teaching classes at the National Conference on Singapore Math Strategies for Grades K-6 with five other American teacher trainers. Several teachers are also sharing their experiences with the participants. I was told that there are 800+ participants from all but four states in the US with California having more than 100 representatives. There are also about 10 international participants from places as diverse as the Uganda, Mexico and the Philippines.

On Monday, I kicked the conference off with a keynote session with Char Forsten. The session is titled Singapore Math: Where East Meet West. I gave the audience the context in which mathematics is taught in Singapore. In particular, I touched on the emphasis on challenging problems in the PSLE, and the fact that most schools have their own problem-solving programme to supplement textbooks which focus on developing strong basics. I also spoke on high parental expectations and LSP and Foundation Mathematics as safety nets to make sure every student get help so that they do not fall too far behind. Many Singapore teachers conduct remedial programme in one form or another - some schools have structured remedial while others do them as and when the need arises. Finally, I touched on teacher education, before they enter the teaching service and when they are in service. I raised the point that perhaps it is possible for teachers to acquire pedagogical content knowledge and content knowledge by working through the student textbook.

Over the four days, I taught classes on a variety of topics ranging from word problems to visualization to lesson study and more. The participants were enthusiatic and appreciative despite the relatively large class size. On the first day, all the three classes I taught (visualization, questioning technique and challenging word problems) had about 200 people.

I will write about these classes. I will upload the slides for the keynote lecture as well as for the different classes at and at

Other than this blog where I talk about the lessons I teach, you may go to for general discussion on Singapore Math. There is a video gallery here. is where I post responses to questions I receive through e-mails. If you are interested in lesson study, go to

Tuesday, June 30, 2009

Some Lesson Ideas for AYG2009

There is a sample lesson plan on incorporating scoring in diving mathematics lessons. See I think scoring in diving is especially useful for Primary 4 topic on operations involving decimals and Primary 5 topic on average.

Saturday, May 23, 2009

NSM101 Lesson on Probability

I taught this lesson in Manila to a group of high school teachers using the New Syllabus Mathematics. The problem posed is to find the probability of drawing two cards with the same number, one from a stack of cards labelled 1 to 4 and another from another stack of cards, also labelled 1 to 4. Using the responses from eight groups, we were able to solve this problem that involved combined events using the knowledge of basic probability of single event. The responses included the use of listing, tree diagrams and table. I was quite satisfied we managed to use the responses to explain the multiplication rule.

Saturday, May 16, 2009

Education in the US

Secretary of Education Arne Duncan
Last week I went to Berkeley County, West Virginia, to begin an open, honest conversation about education reform.I wanted to hear ideas about how we can accomplish President Obama's goal of providing every child in America a complete and competitive education, from cradle through career.As we prepare for the reauthorization of No Child Left Behind, I want to hear from classroom teachers and other educators, parents and students, business people and citizens. What's working, and what's not? What do we need to do that we're not doing, and what do we need to stop doing - or do differently?

In the coming weeks, I will ask questions here. Topics will include raising standards, strengthening teacher quality, using data to improve learning, and turning around low-performing schools. I will be reading what you say. So will others here at the U.S. Department of Education.Today, I want to start with a simple set of questions: Many states in America are independently considering adopting internationally-benchmarked, college and career-ready standards. Is raising standards a good idea? How should we go about it?

Let the conversation begin!
Arne Duncan

Friday, May 15, 2009

Art & Mathematics

Another interesting problems we solve at the Ateneo Summer Institute is the one in which the teachers created interesting pictures like rat, cat and flowers on geoboard. Later, I asked the teachers to predict which of the picture occupies the largest area. Interestingly, in both groups, teachers suggested counting 'dots left over' as an indicator of area. We later tested this theory by finding the area by counting squares.

Some of the pictures were easily dealt with because they were made up of squares and half-squares. Others were more challenging. I hope the teachers got the idea of how to sequence discussion in students' creation in a way they are kept challenged all the time. We moved from cases which were easy to find the area to cases which demanded more visualization.

Monday, May 11, 2009

Ateneo Summer Institute

This is the second year I am teaching this course for early grades teachers (kindergarten to third grade). In today's session, we explore why we teach mathematics. We did a problem that also includes drills-and-practice based on subtraction. Typically subtraction algorithm is done in grades one to three all over the world. I hope the teachers were able to understand that if all the child is able to do is to do subtraction then we have failed. The teachers told me a a cheap calculator cost about 50 pesos retail, that's a little more than USD1. I am sure the wholesale price is even lower. The teachers agreed that a 50-peso calculator can do subtraction rather well. So, if a drill-and-practice exercise enable childrent o do subtraction perfectly then they are just as valuable as a 50-peso calculator. i tried to convey the message that helping children develop thinking and other human competencies are equally important, if not more improtant than developing the ability to perform the subtraction algorithm. A teacher said the activity took more time. I hope we will see that the time invested (50 minutes) is well-spent. In fact, I argue that this is necessary. Unless a task is an extended one, the thinking involved is not very deep and, hence, not significant.
In case you are not in the class today, the problem was to use one set of digit tiles (0 to 9) to make two 4-digit numbers such that the difference is as small as possible. In both classes, we had 25 as the smallest possible difference. The teachers were impressive and were able to make some connections - such as the thousands digits are consecutive. In the end, they found three possible solutions 4xxx - 3xxx = 25, 5xxx-4xxx = 25 and 6xxx - 5xxx = 25. Try it and let me know what other insights you manage to have.
A teacher asked how teachers can formulate such problems. A short answer is to attend courses like these as well as to lookt hem up in books and on the internet. A long answer is to change our beliefs what mathematics teaching is all about. Once that happens, all the tasks we create or choose will have features that allows students to develop good thinking abilities and habits.

Thursday, April 9, 2009

Division of Fractions

I have just completed nine periods of mathematics lesson on this unit with a class of primary five students who struggle with mathematics. I am teaching dividing fraction a/b by a whole number c which is often done using the invert-and-multiply method. Instead of using this method which I believe contribute to these students' difficulties with mathematics, I use an alternate method which based on simple reasoning ... if 4 divided by 4 is 1 then 4 fifths divided by 4 is 1 fifth. In cases when this is not obvious such as 2/3 divided by 4 we talked about changing 2/3 into 4/6. 4 sixths divided by 6 was then "easy beezy" as one student put it. It gave the students plenty of chance to revise equivalent fraction, which is fundamental to operations involving fractions.
I particularly enjoyed the third lesson when I gave each pair of students a fraction. If they thought that their fraction has the same value as the one I put up, they put it next to mine. In other words the board was filled with equivalent fractions which the students can then use to do various division.

Friday, March 13, 2009

e-Lecture on Assessment for Secondary Mathematics

In this lecture, we cover the key ideas in test construction of a mathematics test for secondary students.

It is for pre-service teachers in Singapore.

Research in Mathematics Education

It was an interesting opportunity for me to teach a graduate course at Chiangmai University in March 2009. The 26 students are all studying for a master degree in mathematics education. Some part-time students are doing coursework and need to do an independent study and the rest are full-time students who have to write a thesis. The three-day course aimed to help students with the construction of a research problem.

We spent some time identifying problems related to mathematics education. As the students are all school teachers in primary (grades one to six), middle (grades seven to nine) and high schools (grades ten to twelve), it was not surprising that most of the problems are classroom-related. We spent some time for the students to understand that the research problem is a culmination of their own personal experience and the literature. Their experience with the local situation as well as literature that describes the situation in Thailand (e.g. the TIMSS report) as well as previous studies done in Thailand and elsewhere on related topics are used to generate the content of their research problem.

For example, a student wrote that the primary-level students lack the ability to solve word problems in the topic of time. For example, third graders have difficulty in solving word problems that give the starting time of an activity, the duration of the activity and require students to find the end time of the activity.

This is a problem from the local situation. However, the research problem was still not articulated. For example, the research problem could be to investigate the types of difficulties students have when solving such word problems. A related research problem could be to use the findings from the earlier investigation to design instruction or remediation to help students develop the ability to solve word problems involving time, and to investigate features of instruction or remediation that are able to help students develop this ability.

Wednesday, March 11, 2009

Teaching Mathematics Education Students in Chiangmai University

Why are the students so cheerful in class? Because they are solving simultaneous equation with three unknowns - that's why? It is impossible that students were so cheerful in the mathematics class?
Each student raised a card to their forehead thus was not able to see their own card. Subsequently, each student stated the sum of numbers on their friends' foreheads. Say, A had 7, B had 9 and C had 3. A would have said 12, B 10 and C 16. If the value of the card that A had is a, B is b and C is c then b+c=12, a+c=10 and a+b=16. However, in this case A would have known the values of b and c. The only unknown is a. For secondary students this activity can be used to introduce the idea of solving equations. For primary students, it is a good exercise in addition and subtraction and, more importantly, reasoning.

"Despite resources that are unmatched anywhere in the world, we have let our grades slip, our schools crumble, our teacher quality fall short and other nations outpace us," Obama told the U.S. Hispanic Chamber of Commerce. "The future belongs to the nation that best educates its citizens, and my fellow Americans, we have everything we need to be that nation," he added. The U.S. leader painted the education drive as part of a broader push to promote economic growth in the face of a deep recession and the nation's worst financial crisis in decades.

"In a 21st century world where jobs can be shipped wherever there's an internet connection, where a child born in Dallas is competing with children in Delhi ... education is no longer just a pathway to opportunity and success, it is a prerequisite," he said.
These are words from the President of the USA. In the special lecture I had with all the undergraduate and graduate students of mathematics education in Chiangmai University, I hope the 100 of them in the room can teach 40 students well. In their many years as a teacher of mathematics, they are in the position to bring up a whole generation of citizens who can compete in this type of global economy.

We emphasized on the problem-solving approach, the focus on intellectual competence as well as the concrete/pictorial/abstract approach to teaching students. I was asked where teachers can find problems such as the ones I used in the session. I replied that the problems themselves are not as important as how the teacher uses it. I had demonstrated in the lesson how even computational procedures such as 2/3 divided by 4 can be taught to help students develop and improve their intellectual competence.

In any case there are many good problems on the internet and in moments of inspiration some teachers can come up with interesting problems. If you are one of them, please share na. (na is a particle in Thai to gently persuade someone to do something.)

Sunday, March 1, 2009

Temasek Foundation Singapore & Population and Community Development Association Thailand Project

Think of two numbers e.g. 4 and 5. We will do three steps.
In a variation which I did with second graders in New York, they picked two numbers from a stack of cards. They showed the first number but not the second one. Before that they had done the three steps with the two numbers.

Use the 4 and 5 to make 45. Use the 4 and 5 to make 9 (4 + 5). Finally find the difference between 45 and 9.

After demonstrating that if they tell me the first number then I will be able to read their mind for the second and do all the three steps and giving the right answer each time, we began to record the findings.

Many of the participants saw the multiple of nine trick. As I pressed for more observations, another interesting one is how the second digit in the answer is derived by subtracting the first number from ten.

I was reminded again that asking students to explore more ways to often result interesting and outstanding responses.

I did this problem with the teachers group as well as the principals group and on both days we got equally interesting responses except I was surprised that no one in the teachers asked for the reason why the trick worked. One did in the pricipals group.

Friday, February 27, 2009

Temasek Foundation Singapore & Population and Community Development Association Thailand Project

In this Temasek Foundation funded project, Population and Community Development Association (PDA) in Thailand has put up a Train The Trainers (TOT) Project. The idea is that teachers and principals are trained in the curriulum and management system used at the Lamplaimaat Pattana School in Buriram, a school that has a track record on par with that of private schools in the nation. The one-day training on effective teaching and learning of mathematics served as a case study to help teachers think about curriculum implementation and for principals to think about how they can sustain and implement high-quality programmes.

It was encouraging too see even the school principals thinking hard to come up with the solutions to the problems posed to them. And did they have good and interesting ways of solving those mathematics problems! I hope the general message about shifting the focus from memorizing fatcs, carrying out procedures without understanding and doing tedious clerical tasks like computations to intellectual tasks like doing visualization, looking for connections, developing number sense, developing metacognition and enhancing communication came across during the seminar.
There were 200 participants on each of the two days. So, I was glad to have the opportunity to influence the thinking of 400 people who are in the position to do something to make the learning process in schools effective and enjoyable. There were also three teacher trainers from a teacher training institute in Luang Prabang, Laos.

Saturday, February 21, 2009

Teacher Workshop @ Queenstown Primary School

The workshop was on developing visualization in primary mathematics. An interesting point raised by one of the teachers is whether it was easy for students to visualize. It set me thinking and I then suggested that visualization may well has a cognitive component as well as an affective component. Another way to put it is that it has a skill component - can visualize or not - as well as a habit component - is the mind in the habit of seeing the same thing in different ways or not. Thus, can students see that 2/3 dividied by 4 is the same as 1/4 of 2/3, 1/2 of 1/3 as well as 1/6 of the whole depends both on the possession of the skill to interpret diagrams as well as the habit of being able to interpret the same diagram in different ways.

Thursday, January 29, 2009

DCM201 MicroTeaching on Subtraction of Mixed Numbers

It is promising that I see at least one very good lesson in every microteaching session. teaching is a complex activity that defies description in words. one has to see it being played out before one's eyes. I hope the trainee teachers in my group who are observers are capable of seeing good examples of instruction. This is why I assign them the roles of 'students' and 'observers'. On the days they are the 'observers' they can divert all their attention to witness the complex activity of teaching without being distracted.

In Rendra's ;esson, we saw the use of PowerPoint to model the subtraction process. I wished he had stuck to this. Unfortunately, he moved to doing the subtraction in abstract symbols too soon. In Naz lesson, which was brilliant, I saw him being able to handle the complexities of being a teacher bringing a lesson plan to life. In particular, the pictorial-abstract linkage was well done. He used the 'models' to explain the symbolic manipulation. The subtraction process can be made obvious using suitable straegies that trainee teachers should be able figure out. In Hafeez' lesson, the idea of regrouing was not modelled by the pictorial representation. The use of a strory was good. especially if the teacher can act the situations out by making voices for Mr Gopan and whoever else in the story. Also, the pizza was impressive,

Monday, January 26, 2009

DCM202 Challenging Problems (1)

The problem is to find the tens digit when 2 is multiplied 2009 times. Most of the students were able to solve simpler problems to see a pattern. A strategy suggested was to try simpler cases until we get the same digit for the tens and ones digit. I was glad I gave the students time to establish the fact that the pattern will definitely repeat. That means they were expected to be able to explain the pattern. I felt that it was insufficient for students to solve the problem based on pattern observation alone. They should be able to show that the pattern is guaranteed.

Thursday, January 22, 2009

DCM201 Division of Small Numbers

Keng Sing's lesson using When The Door Bell Rang was enjoyable with cookies and plates for the children to act division out as sharing. It could be improved by applying the theory of variation. In all the examples the children were asked to act the situation out. In the later stages, the children could be asked to draw and use the cookies to confirm their ideas.

Rebecca's use of the cut-outs was suitable. I have suggested that the writing of the division sentence happens sooner in the lesson. And that was what Wei Wen did. In her lesson on division with remainder, she made careful links between the concrete with the pictorial and abstract represents well. One suggestion is to introduce the idea of quotient a little later.

Tuesday, January 20, 2009

DCM201 Multiplication of Large Numbers

Multiplication of large numbers begin in Primary 3 with multiplication of a 2 digit or 3-digit number and a 1-digit number followed by that of a 4-digit number and a 1-digit number. Later, students learn how to multiply a 2-digit number with another 2-digit number. The last two are often taught in Primary 4.

Khai taught 3D x 1D. I am pleased that he used the base ten blocks to model the process of 231 x 3 as representing 231 as 200, 30 and 1. Later he modelled the multiplication process of the hundreds, tens and ones. It is satisfying to see part of Bruner's Theory in action. One way this lesson could be improved, I told Khai, is to separates cases that requires renaming (365 x 5) from those that do not (231 x 3) and teach them in separate lessons.

In another lesson of 4D x 1D, I was excited when I saw the teacher teaching children how to use 123 x 2 to do 2123 x 2. The simplicity of it was beautiful - the new lesson simplify requires Primary 4 students to do the multiplication of the thousands (2000 x 2) and add this to 123 x 2 (which was done the year before already). It would have been great if this was done for the entire lesson. In this lesson it was done too briefly for it to have much impact on children learning. A good example of Teach Less Learn More (students learn to make connections by linking 123 x 2 to 2123 x 2).

In the third lesson, Lawrence did an excellent job applying Bruner's Theory in linking concrete, pictorial and abstract representations of 2D x 2D. You got to be there to appreciate fully what I said but breaking 34 x 42 into 4 x 42 and 30 x 42 using the area model would help children develop visualization as they learn the algorithm.

It was a good session. All the three demonstrates the basic competencies of competent teachers. They were the first three, so you can understand the nerves. It was good to see how the rest of the students who acted as students and as observers in a 'pseudo' lesson study supported their colleagues. I look forward to 27 more lessons that will unfold in the next few weeks preceeding the practicum.

Friday, January 9, 2009

Stamford Primary School

This was a workshop on mathematical problem solving. We spent most of the time on undertanding the importance of number sense, looking for patterns, articulation of thought and visualization. We also explores two ways to improve understanding of word problems - metacognitive reading and problem posing. Finally, we explore the what-if strategy as a post-solving strategy. In one of the problems, teachers are asked to look for relationship between and among a set of numbers generated in a mind-reading game. The idea is to tell kids that if they think of two digits (say, 5 and 2) and tell you only the first one, then you can rread their mind to get the secon digit and even able to use the digits to (a) form a number 52, (b) form another number 7, from 5 + 2 and finally (c) find the difference between 52 and 7 and be able to write that down on a piece of paper. All these without them having to tell you the second digit. So when the kid tells you 5, you can write 45 down. When the kid tells you 8, you write 72 and when the kid says 9, you write 81. What do you write on the piece of paper if the kid says 3. 27 is correct. How does it work. That was the problem the teachers solved. They came up with two strategies (1) multiply the number the kid says by 9 and (2) subtract 1 from the number the kid says and then making sure the sum of the digits in the answer is 9. Do you know why the trick works? Using this problem, teachers can see that problem solving involves the ability to see a pattern and generalize the pattern. Also, it involves putting our ideas into words as we try to describe our observations of how the three numbers are related.
This workshop was attended by n teachers.

Thursday, January 8, 2009

DCM202 Mathematics Games

When your friend says Salute, each player draws a card and outs it on his or her forehead. That means you can see everyone's card but not your own. Each player tells the sum of all the numbers he or she can see. The object of the game is to tell what number is on the card on your own forehead! After playing the game, did you realise what this game can do to help your sudents learn? Did you realise the mathematics that students can learn while playing this game? It was good to see the smiles and hear the laughter during this and other games that you played.

Throuhout the week, the students played several other games. I was hoping that they can abstract for themselves what roles games can play in the learning process. There was this game where they had to piece together 9 squares to form a larger square in such a way that the adjacent numbers are equal.

There was also this game of I Have... Who Has? where players have cards that read I Have 40. Who has 10% of 280? and another player who has it will continue I Have 28? Who has 5% of 280?. Once you have read from the card, you can get rid of it. The aim of the game is to lose all your cards.

The approach I adopted was to let the student teachers play the game so that they can abstract for themselves the importance of games in the learning process. Is it necessary for me to put in some structures so that this abstraction has a higher chance of happening. I decided on this approach as opposed to giving them a lecture to describe the different ways to use games - for drills, for developing concepts, for problem solving, for motivation and so on - and illustrated by specific examples because I want them to engage in the games so that they will remember the games and think of using them when they are in schools.

DCM202 The Role of Calculators in Primary Mathematics

This is the first lesson with the two classes who offer this course. I have planned to do some calculator activities with the teachers and get them to reflect on the roles of the calculator in the primary mathematics classroom. The students were asked to find a pair of two-digit numbers such that AB x CD = BA x DC. As expected, solutions such as 11 x 33 and 12 x 21 came quickly. Later, solutions using three or four distict digits such as 24 x 84 and 63 x 12 were given. One class gave nine different solutions while the other gave eight. As usual the students observed relationship between the product of the tens digits and the product of the ones digit (such as 2 x 8 and 4 x 4 in 24 x 84 and 6 x 1 and 3 x 2 in 63 x 12). In 93 x 26, students observed the relationship between the ratio 9/6 and 3/2. It is good that some students could say that these two are essentially the same. It shows that they were making linkages. It was particularly satisfying when one student gave an incorrect answer 41 x 22 (which I wrote down without judgement) and another group of students using that as a basis to obtain a correct answer 42 x 12. I was thrilled when another student came up with a novel observation that in 24 x 84, doubling the 24 gives 48 and reversing the 48 gives 84 - which is the larger number. Similarly with 42 x 12, doubling 12 gives 24, reversing the digits in 24 gives 48 - and, hey, that's the larger number. I was so excited by this that I forgot to tell the class that this is a rare observation. It could well be the first time I hear it although I have done this lesson countless number of time.

At the end of the lesson, I was wondering if my students could see the role of the calculator in this activity. And how the role of the calculator in this activity is different from its role in the other activities that we did that day and in subsequent lessons.