Fractions

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

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

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

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

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

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

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.