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

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.