A Bunch of Questions

This is my response to a set of interview question by a parenting magazine.

👀How can I help my child nurture a love for Mathematics?

😜Children who grow up having a love for mathematics are those who have cultivated a productive mindset about what mathematics is. 

You are unlikely to enjoy mathematics if you think that it is all about carrying out rote procedures, memorising and doing tedious computations. 

You are likely to love mathematics if you see it as a field where you figure things out and one where you see patterns.

Children who learn mathematics by manipulating concrete materials and using diagram, children who learn mathematics by interacting with other people, children who learn mathematics without being burdened by jargons right from the start, children who learn mathematics by constructing meaning for themselves, these are children who would love the subject.

👀What materials can I use to facilitate my child's understanding of Heuristics?

😜What does the word 'heuristics' mean to you? A heuristic is simply a way to solve problems. And there are many ways - act it out, draw a diagram, make a guess, write an equations are some examples of heuristics. Heuristics are rule of thumb that one use, often in combinations, to progressively move towards solutions of problems. How do adults facilitate children's understanding of heuristics? By choosing problems that lend themselves to the use of specific heuristics you want the child to learn and by letting them use the heuristics in their own way. Do not place too many constraints on the way they use them. Heuristics are pretty flexible and there is no one way to do, say, guess and check.

👀How often should my child practise math modelling so that he can be proficient in it? 

😜I think you mean drawing bar models. Mathematical modelling is more than just bar models. Children should always be encouraged to represent information using diagrams. They develop visualization when they do so. So, the answer is as often as possible.

👀How can I make Mathematics more relatable to real life situations?

😜That's easy because mathematics was created to describe and extend the world around us. You see multiplication when you see cookies on a tray. You create art by arranging shapes in a tangram set. Dice used in by rad games give rise to all sorts of mathematics problems. Quadratic equations describe the path of a ball thrown in the playground. Tennis tournaments can be described by exponential function. Our everyday language is peppered with mathematical ideas - you often say success is a function of hard work. 

👀How do I handle my child's frustrations in being unable to solve Mathematics problems?

😜By not spoon-feeding them right from the beginning. Children who are independent are resilient and not easily frustrated when faced with a problem they cannot solve straight away. 

Children who are frustrated easily when they cannot solve a problem tend to see mathematics as something that involves routines tasks which can be solved quickly using a formula. 

Find out the reason why they cannot solve a problem - is it the comprehension of the problem, is it the inability in handling multi step situations, is it the calculation? And provide help in that area.

👀How can I help/support my child if he is lagging behind in the classroom? 

😜Focus on basic skills and tasks that are routine.

Explore alternate ways - if long division (say 351 divided by 3) is troubling them, get them to see that 351 = 300 and 30 and 21, all of which are easily divisible by 3.

Use diagrams to help them visualize - it is much easier to see two-thirds of three-fourths using a bar model.

Avoid explaining solutions to children who are struggling to grasp ideas. Scaffold their learning by asking questions. Concrete materials and diagrams are always helpful.

👀Should I engage my child in collaborative learning (eg. learning circles) outside of school?

😜Collaborative learning is always good.

Question about Student Support and Teacher Education

Thanks for your presentation at the Blake School on Monday afternoon.  I was attending with a team of math intervention teachers from our school district in Edina, MN, and we have some questions about differentiation.

1. In Singapore, how do schools provide support for students that are not keeping up with the rest of the class?

2. How do schools support students that are consistently one or two grade levels ahead of their classmates?

Additionally, do most of Singapore’s teachers matriculate from the same university? If so, do you think that is a benefit to math education or a detriment?

I’m looking forward to seeing you again this afternoon in Wayzata!

BanHar responds ...

All Singapore teachers complete their pre-service education from the same university (National Institute of Education NIE). Plus, it ensures consistency but we need to make sure this place provide the best teacher education money can buy. On the down side, the institution faces no competition and unless it has the discipline to improve and innovate despite the lack of competition, it can be the start of a decline. I hope our NIE is of the former. (I used to teach there.)

1. In Grade 1, students who are not on grade level competency receives support in mathematics and English Language - it is a pullout (we call it learning support program in mathematics or LSM). At all other levels, they receive extra support during class (in more difficult cases, there is a learning support teacher in class but that is not the norm). Usually, the teacher will have to take care of these students when the others are doing independent work. In many cases, we will meet with them after school for an hour or so once a week for remedial work. Having said all that, the use of concrete materials is to help every learner, including the struggling ones, learn well.

So we have in-class as well as pullout / separate remedial classes. In some cases we have a learning support teacher ( we call them allied educators, they are not trained teachers but undergo a training program for the job).

2. We provide these students with more challenging problems i.e. the enrichment approach. We do not accelerate them. However, these challenge is also made available to all other students. At upper levels (grades five on wards but sometimes as early as grade three, we track these students i.e. they are put into a separate class and the teachers do more challenging stuff with them, again not accelerating but provide more challenging tasks). For the most advanced they join the so-called gifted education program (GEP) from grade four onwards, where they are challenged with all kinds of things include research and project. From grade seven there are schools that they can go to to enhance their ability / interest e.g. schools of science and mathematics etc. 

Lesson Study Question

I am currently leading my school's (primary) lesson study efforts. 

I would just like to seek your expert advice on the number of cycles of Lessons Study. Right now, my school currently embarks on two lesson study cycles. Some teachers have actually asked if the second cycle itself can be done not by observing just one teacher, but by taking the improved lesson plan back to their classes and conducting it. 

Would such a practice dilute the essence of lesson study given that there will thus be only one lesson during the first cycle where teachers observe teaching and understanding? 

I would greatly appreciate your expertise and take on this subject matter. 

Brenda, Singapore

You can have teachers taking the lesson and then teaching it to their own classes after the lesson refinement stage. In this case, the team has completed one cycle of the lesson. It is fine. Of course, going into the second cycle offers teachers with another opportunity to observe and talk about the lesson. If the team is motivated and can afford the time, doing a second cycle should offer them new things to see and should enrich the professional learning.

Pies and Tarts

I am  a parent of a student in the school you gave a seminar. I found a problem in an assessment book which was confusing. Can you please help me to find out a solution to that problem?

I do not understand the way the answer given in the assessment book.
A Parent in Singapore

The Problem

Students with good number sense will know that the number of tarts and pies have to be in certain multiples given that they do not come in halves and quarters in a shop.

The ratio 3 : 2 tells us that tarts are in multiples of 3 (and pies in multiples of two) but as the tarts are packed in boxes of 4's, it must be in a common multiple of 3 and 4 (12). In the same way, the pies come in common multiple of 2 and 3 (6).

The least possible number for this to happen is 36 tarts and 24 pies - 9 boxes of 4 tarts and 8 boxes of pies. These will be $9 x $5 and 8 x $4 or $45 + $32 = $77.

It is also given that if all the tarts and pies are sold then they will bring in $770.

I think you can finish this up now.

This problem is based on ratio (P5), multiples (P4) and some basic multiplication and division (P3) and addition (P1).

This is a challenging P5 or P6 problem.

Note: P5 = Grade 5

Skeleton of the Solution

About Functions

Can you guide me how to solve Question 7b?

This is from iGCSE past examination paper. I could not even start.

Given that f(x) = 10^x.
(a) Calculate f(0.5).
(b) Write down the value of f^-1(1).

Usman

f^-1(x) is the inverse of function f.
Given that function f(x) = 10^x
So, f(1) = 10^1 = 10
f(2) = 10^2 = (10)(10) = 100 and so on
I think you can figure out 10^(0.5) ... it is the square root of ten, isn't it?

Let's get to 7(b), which is the one you wanted help in.

First you need to know the meaning of inverse of a function.
If a function is 2x (doubling a number) then its inverse is (1/2)x i.e. halving the number.

Given that f(x) = 10^x, the second part is asking you what is the value of x when f(x) = 1.
The answer is x = 0 because 10^0 = 1.

(Note: I use ^ to mean to the power of)

You may want to review the idea of inverse function.

About Range

I am a fourth grade teacher and we are currently working with mean, median, mode and range.  I received a parent email asking to clarify the definition of range.  Her email is as follows:

“ A question came up during homework on the definition of range. Apparently the kids' textbook says that the "range is the difference between the least number and the greatest number".  Range = greatest value - least value. That is not the correct definition. Range is the least number to the greatest. For example, if the test scores in one class range are between 90 to 100, and another class between 60-70. The textbook definition would say the range for scores in both classes is 10, which obvious does not make sense. (I think their textbook is defining what is known as the span of the data but that is rarely used.) A number of parents were puzzled about it but we don't know if this needs to be corrected or at this level the textbook definition should stand. Would you please clarify.”

Am I correct that finding the range does require students to subtract the greatest from the least amount in the data set?  I am not sure how to respond to her question.

Jillian, New York

Definition of Range: 

There are many places one can check on the definition - wikipedia quoted reliable mathematics / statistics textbooks. This is another source for a definition.

You can send the parent some of these links. 

We are often interested about the 'average' in a data set as well how the data distributes itself around the average. Examples of average include mean and median. examples of a measure of this distribution includes range and standard deviation.

Range is the size of the smallest interval that contains all the data and tells us about statistical dispersion.

I tried checking what the parent referred to as "span of the data" but could not find any entry on the internet. Apparently, it is not a conventional term. Range is a more conventional term to describe the idea under discussion. 

Question on Division Bar Model

Trish from Hawaii asked a question about setting up the bar model in a division type word problem.

Question by Trish (Hawaii) from Jimmy Keng

Of course it is not necessary to solve every problem using the bar model.

Spiral Curriculum

Good day Dr. Yeap! I attended the seminar last May 21-24 at SM Megall (see photo). Thank you for sharing your expertise and time with us. I just want to ask you about spiral progression. How do you apply it in Math? Can you give an example? Cely

Singapore mathematics curriculum emphasizes the spiral approach based on Jerome Bruner's explanation on spiral curriculum. The idea of the spiral curriculum, according to Bruner - 'A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them.' Most people will not miss the idea of 'repeatedly' but may miss the subtle notion of 'building upon them' and 'until the student has grasped the full formal apparatus' of the target concept. In Singapore curriculum, addition is taught four times in Grade 1 (this is a core idea and they are new to it) - addition with 10, within 20, within 40 and within 100. Students get to revisit the idea of addition repeatedly but each time building on the strategies that they already had. When they add with 10, they count all and count on, perhaps with the use of concrete objects and drawings. Later, in addition within 20, they learn to make ten before adding, effectively acquiring the notion of place value. Later they progress to more formal approaches such as adding ones and adding tens in the formal algorithm. Thus, it is not mere review of materials. It involves extension. In a similar way, multiplication of whole numbers is taught in grades one through four; addition and subtraction of fractions is taught in grades two through five; area of plane figures is taught in grades three through seven; solving equations is taught in grades seven through nine. As a result, in Singapore, Algebra is taught across grade levels in high school (grades seven through twelve). Thus, we do not have the practice of teaching Algebra, Geometry etc separately. They are all under the subject of mathematics. There is geometry in all grade levels.

Question on Retention

Question - How do we help students retain what they have learned.

ATeacher in Hawaii

Answer

A good program arranges the topics in a certain way for various purposes. To help students retain, one of the ways is to arrange topics in such as a way that students have many opportunities to revisit a particular key concept or skill.

Let's use the example of equivalent fractions. After it is learned in Grade 3, the chapters that follow it are adding unlike fractions and subtracting unlike fractions (but limited to cases such as a third plus a sixth or three tenths subtract a fifth where it is necessary to rename only one of the fractions to make both like fractions). The two weeks or so of constant writing fractions as equivalent ones as students add unlike fractions help them consolidate the skill learned. It is critical that a new skill is well consolidated before students leave them behind.

The next grade level when student said such fractions but for cases where the sum exceeds 1, students get to review finding equivalent fractions. In grade five, when students are dividing say a half by three, one of the methods involves renaming one half as three sixths before proceeding to divide the three parts into three. Using this method, three fourths divided by three can be done straight away but to divide three fourths by two requires renaming three fourths as six eighths. The opportunity to review equivalent fraction in a different context enhance the retention of the skill of finding equivalent fractions.

Two principles discussed here - ample consolidation after a new skill is learned and review but not a mere repetition but done in other / more challenging contexts.

Singapore curriculum is arranged for this to happen. Good textbook authors arrange the topics for this to happen.

A Bunch of Questions

Question
How can Singapore Math be taught to children who are at different learning levels? Answer
We just completed a two-day professional development on how to do differentiated instruction using Singapore Math in White Plains with 60 teachers. For struggling learners the concrete experience before pictorial representation helps them. For advanced learners, tasks can be easily extended to engage them in higher-order thinking. Singapore Math is well-known for helping average learners reach high levels of achievement. The last piece of information is what emerges from TIMSS and PISA where many of our average learners are performing at Advanced level in TIMSS or Levels 5 and 6 at PISA.

Question
How does Singapore Math help children who have difficulty learning math?
Answer
The use of visuals helps. In Singapore Math we use the CPA (concrete-pictorial-abstract) Approach based on Jerome Bruner's work. Students are taught an abstract concept via concrete expreinecs and the use of pictorial representations.

Question
How does Singapore Math help children with more advanced math skills? Singapore Math is based on the idea of using mathematics as a vehicle for the development and improvement of a person's intellectual competencies. Students with advanced skills get to work with more complex problem that requires deeper thinking (this is in the program). They also get to develop skills such visualization and ability to see patterns.

Question
What training do teachers/school administrators need in order to introduce Singapore Math in their curriculum?
Answer
Marshall Cavendish Institute offers a range of professional development courses to help teachers teach the program. MCI's Experiencing Singapore Math is a one-day executive program for administrators who are keen to implement the program in their schools. The publisher HMH has ran this program in Chicago, Nashville, Alabama, Scottsdale, and recently in Neward and East Brunswick in NJ.

Question
What grade levels seem to show the greatest impact for Singapore Math?
Answer
I do not know this but you may refer to various research done at various sites. Based on my professional experience, best results come when the kids are introduced to it at K-2.

Question about Students Not Meeting the Goals

Question:
What if a student is clearly not ready to move on to the next level at the end of the school year despite concerted efforts to assist him/her? What do you folks do with these students (grade K-6)?

Malama pono (Take care),
Teacher in a oublic charter school in Hawai'i

Answer:
coming soon

Question on Learning Theories

Last week I attended your lecture in Vlissingen where you spoke about the Singapore approach. During the lecture you mentioned some names of people who wrote about the CPA approach, the Conceptual approach and about systematic variation in your powerpoint presentation. I'm very interested in reading articles about these approaches.


Educator in the Netherlands

You can goggle Jerome Bruner (spiral curriculum, enactive-iconic-symbolic representations), Zoltan Dienes (variability) and Richard Skemp (relational understanding, instrumental understanding).

These are the key theories taught at the NIE to Singapore mathematics teachers.

Questions at AMSTI Math Leadership Symposium

Do primary school teachers in Singapore teach more than one subject?

Singapore primary teachers are trained to teach more than one subject (two or three subjects). See details at National Institute of Education. Typically, a teacher teaches more than one subjects, especially at lower grade levels. In some cases, especially at higher grade levels some teachers may teach multiple classes of the same subject. Teachers teaching Mother Tongue, Music, Art and Physical Education are more likely to be teaching only one subject. It is only at secondary level that teachers specialize somewhat (two teaching subjects).

Other than English and Mathematics which are core subjects other subjects include Science, Mother Tongue Language as well as non-examinable subjects like Art, Music, P.E., Social Studies, Health Education, Civics and Moral Education and a host of other programs such as Porgram for Active Learning (PAL).

More to come

Textbooks in JC

I am a student self-studying mathematics from Singapore's textbook at the 10th to 11th (Secondary 5) grade levels. When I finish these, I would like to continue following Singapore's curriculum. Which books (series) do you recommend. I plan to study advanced topics in depth, like statistics and calculus? I would like to follow the newest books, ones followed currently in Singapore by JC students.

Student in USA

In JC, students do not use a textbook because the lecturers provide their own lecture notes, like in the university. If you have not done Additional Mathematics, you may work on this - it is a book that includes trigonometry, calculus and other advanced topics. This is used in Grade 9 - 10 by advanced students. It is available from the same place you got your other books.

Percent

I hope you still remember me. Last time I contacted you regarding congruence and similarity. Now, my son is studying in Grade 8.

If you don't mind can you clear my confusion regarding one problem of percentage. I have difference of opinion with the mathematics teacher and I want to clear my concept. The problem is as follows:

When we are converting 2/5 to percentage we write as follows:

2/5 x 100% = 40%. ---------- (1)

The teacher says it is not necassary to write symbol(%) with 100.

2/5 x 100 = 40%. ------------(2)

You will notice that % is missing from (2). Can you please explain is it also the correct practice not to write the symbol % with 100. My view is that 2/5 x 100 will result in 40 and not 40%.

I am afraid that this mistake will results in marks being deducted in the IGCSE.

A father in Saudi Arabia


You are right 2/5 is equal to 2/ 5 x 1 and 1 = 100/100 which is written as 100%.

2/5 x 100 = 40 as you said. 40% is equal to 0.4 (not 40).

Perhaps the teacher knows that, in a lenient way or marking, candidates get full credit whether they did (1) or (2). But I am sure you rather your son learn what is mathematically correct rather what is minimally acceptable to earn a credit in the examination. Please advise your son to write the mathematically correct sentence. I also trust that you will help him understand why (2) is not correct (although it may still earn a credit in the examinations, according to the teacher).

From a Dutch Homeschooling Parent

I'm a homeschooling mother from the Netherlands and I have used (and still use) Singapore Math for my children in all the years of their primary education. I can't rave enough about this method of math education: the books are great, the bar diagrams are marvellous - I wish I had learned mathematics this way in my time.

Now my oldest child is starting Secondary Education the upcoming year. In the Netherlands secondary education consists of 6 years and the level of math at the final exams is relatively high (higher than say, American High School level - to give comparison). However, instead of preparing my children for their final exams with Dutch secondary math books, I prefer to keep on working with Singapore Math. Even though I can't really compare Dutch and Singapore High School Math, I very much like the way Singapore Math has build a strong math education in my children in such a thorough and painless way.

But in my search for information I got a little confused about all the available series. There seem to be:

1) New Mathematics Counts Series
2) New Elementary Maths Series and
3) New Syllabus Mathematics

Would you be able to explain to me the difference between these series or can you give me an advice on what to use for the High School years of my children? I'm aware this is not the type of problem question you usually receive on your blog, but I very much hope you can answer me just the same.

For the primary years we're using the My Pals are Here Maths Series (Grades 1-6), and I'd like use the best available SM high school sequence. I prefer to use the series that is used in most Singapore and/or International High Schools. For example: I understand that for the primary years My Pals are Here Maths and Shaping Maths are the most common used series in Singapore and International schools: MPAH in 80% of the schools, Shaping Maths in 20% - roughly estimated.

Do you happen to know the ratio of the three above mentioned series for the secondary years (New Mathematics Counts Series / New Elementary Maths Series / New Syllabus Mathematics)? Or do you know any distinctive features?

It seems that all Singapore Math distributors, either in the US, UK or Singapore are all pushing the series that offers them the most profit, so I don't know who to believe in that area. Buying all the books for four children is quite an investment, so it would be a pity if I invested in the wrong series. I was even on the verge of abandoning Singapore Math for a curriculum called The Art of Problem Solving (http://www.artofproblemsolving.com/Store/curriculum.php) because I didn't have any clue of the right series. But then I found your blog, I hope you can find the time to answer me.

Thanks in advance for your effort.

A Parent in The Netherlands

In Singapore, some students study 4 years and others 5 years for their secondary education. They then move to another two years in junior college (some opt to go to a polytechnic instead). The first four years leads to GCE O Levels (Grades 7-10) and the last two years leads to GCE A Levels (Grades 11-12).

You are right - there are many textbook series for secondary levels. In fact, more than the three you listed. Essentially they are all the same. Internationally, most schools / parents use New Syllabus Mathematics (NSM) and New Elementary Mathematics (NEM) simply because these has been around for many years. NSM is still sued in Singapore schools. NEM is no longer used - the publisher sometimes did not submit them for review or update it to make it a 100% fit with topics in the revised curriculum. All our textbooks must be reviewed by Ministry of Education. In my opinion, NEM is of good quality too.

New Mathematics Counts (NMC) is designed for academically weaker students - the program is to be done over five, instead of four years.

NSM is designed for students who have a strong foundation in mathematics. That is why other than the four books, there is a fifth book that advanced students opt for in Grade 9/10 (they use this book over two year to supplement the main text). This is Additional Mathematics. Many kids in Singapore study this subject. By the time they complete Additional Mathematics, they would have done basic calculus. Generally, if your child do well using the Singapore textbooks, he/she should be ready for ay kind of test.

In brief, NMC if the child is struggling with math. NEM or NSM if the student is quite good in math. NSM has the option for advanced topics.

See http://atl.moe.gov.sg/ for textbooks used in Singapore. Titles with Sec 5N are designed for students who tend to struggle somewhat with mathematics - the topics coverage is the same but done over five instead of four years. Books listed as NT (Nornal Technical) are for students who are moving to vocational course after Grade 10.

Bar Models

This morning I taught Primary 4 students word problems using bar models. I expected the students to get a clear picture by it, but actually not. They were so impatient to compute directly, even without reading the question :( They just depended on my instruction whether to multiply or to divide. So, I'm thinking of an activity at the beginning to introduce the bar model to them - perhaps by making the bars using color papers.

Teacher in Indonesia

This is not unexpected if the students are already used to a computational approach. All they want to do is to compute. Without a clear understanding of the problem, they will not be able to identify the correct operations.

Bar modelling begins in kindergarten when a teacher models the story using the real things and later pictures of the real thing. Later, 5 unifix cubes / snap cubes are used to represent, say 5 sweets. By Primary 2, they begin to use a bar to represent quantities.

Your idea of using paper strips is excellent. I do it all the time.

You can also give problems without numbers.

Johan has more sweets than Siti.
How many sweets does Siti have?

After students show the correct bars for number of sweets Johan has (longer) and number of sweets Siti has (shorter), tell them Johan has 6 sweets more than Siti. Ask them to put in the new information on the bar models. Finally tell them that Johan has 14 sweets. Get them to solve the problem.

Hope this helps.

I have attached a picture of a P3 Maths Question. The answer in blue is the student's answer.

My first thought was that the boy had given the correct answer. However, another parent pointed out that the question could mean having 8 sets of 2 square tables.

We would appreciate your view on this.

A Singapore Parent

I think the boy is correct. The test item would have been clearer if it is phrased this way:

A long table is formed by placing square tables side-by-side. Each side of the square table seats one person.

(This is followed by diagrams of two square tables, three square table and four square tables placed side-by-side.)

How many persons can the long table seat when 8 square tables are placed side-by-side?

I can understand why some adults may think that the answer is 34. They are thinking (because of the diagram) that a table is formed using two square tables and take the question to be asking for the situations when 8 such two-unit tables are used. This issue can be resolved by making the task clearer. One way to do so is suggested.

Graduate Research on Singapore Math

I am a Prep teacher in a school in the Philippines. Our school has adapted Singapore Math for the past four years. I am currently doing a paper on Singapore Math at a university in Manila.

I wish to ask you if there are any materials, readings, or websites that you can recommend?

Best,

Ms. Patricia

Dear Patricia

Actually Singapore Math is based on learning theories. If you are doing a paper on Singapore Math you should be reading Bruner's theories on representations and spiral curriculum, also on Piaget's ideas on how children learn, Dienes theory of variability, Skemp's ideas on relational and instrumental understanding, and show how Singapore Math is consistent with learning theories.

If you are doing empirical research, there is a large number of areas you can research on e.g. how concrete materials help Prep children progress to pictorial and symbolic representations.

I will be happy to read your final paper.

Currently, I am teaching the topic on time.

We are beginning to teach pupils how to draw timeline instead of writing 2.30 pm + 30 mins = 3 p.m. I have reminded my pupils why 2.30 pm + 30 mins = 3 p.m. because a specific time cannot be added to duration. Despite going through in detail in class, I was surprised that some of my pupils were still writing the number sentence 2.30 p.m. + 30 mins = 3 p.m. though they drew the number line as well.

Moreover, their number line were not very helpful. It is obvious that that they had worked out the answers mentally in their heads.

Why is my pupils still writing it incorrectly?

Primary Three Teacher

This could be due to some of them being given extra (incorrect) instruction at home. The fact that the incorrect number number sentence give them the correct answer does not help your cause either.

It is good you tell them it is wrong. But be prepared for some who will take a longer time to represent the ideas correctly. Continue with the time line representation.

I am glad you pointed out that a point in time (e.g. 2 p.m.) cannot be added to time duration (e.g. 2 hours). Some students are taught by adults who make this mistakes themselves.