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:

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 ( 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.