Tuesday, August 18, 2009

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

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%.

Thursday, July 23, 2009

Foundation Mathematics

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.

Saturday, July 18, 2009

Language Issue

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

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

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.


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).

Wednesday, July 1, 2009

Model Method

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?

Thursday, June 18, 2009

A Multiple Problem

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?

Thursday, June 4, 2009

Pre-School Numeracy & Assessment

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.
AnamarĂ­a, 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.

Wednesday, June 3, 2009

Test Items from Singapore National Tests

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).

Monday, May 25, 2009

Word Problem

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.

Monday, May 11, 2009

Speed Problem...Again

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?
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.

Friday, May 8, 2009

Request for Presentation Slides

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.

Monday, April 27, 2009

A Problem from a Hong Kong School

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?

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.

Monday, April 13, 2009

Another Speed Problem

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

Saturday, April 11, 2009

Speed Problem

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.
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.

Friday, April 10, 2009

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.
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.

Thursday, April 9, 2009

Goggles Problem

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.

Sunday, April 5, 2009

Classroom Management

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

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!
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?

Friday, April 3, 2009

Word Problem on Speed

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?

Wednesday, April 1, 2009

Freudenthal Institute

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


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

Monday, March 23, 2009

Base Ten Blocks

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.

Tuesday, March 17, 2009

Fraction of a Fraction Word Problems

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.

Friday, March 13, 2009

Research Network

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

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?

Friday, March 6, 2009


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

Thursday, March 5, 2009

Multiplication & Division Algorithms

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.

Tuesday, March 3, 2009

Learning Support

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

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.

Friday, February 27, 2009

Number Patterns

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.

Monday, February 23, 2009


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

Thursday, February 12, 2009

Division of Decimals & Mixed Numbers

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.

Saturday, February 7, 2009

Number Bonds

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.