*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".*

## Tuesday, August 18, 2009

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

## 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*

## 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*

### 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

### 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.*

## Wednesday, July 1, 2009

### Model Method

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

*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 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.

*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

*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

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

*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

*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

AB + B = BA

A = ________B = ________

How do you teach a P1 child algebra?

Lily

*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

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.

Car A travels 20km more than Car B in an hour. (80-60)

## Saturday, April 11, 2009

### Speed Problem

*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

*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?*

*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

*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

*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

*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

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

http://www.utrechtsummerschool.nl/

http://www.science.uu.nl/summerschools

http://www.fi.uu.nl/fisme/en/

http://www.science.uu.nl/summerschools/appliedsciences/

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

## Monday, March 23, 2009

### Base Ten Blocks

*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.*

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?*

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

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

### Research

*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

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

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

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

*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:

*You should congratulate your child. He has done a good job.*

## Monday, February 23, 2009

### Research

*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

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

*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.*