Wait, math from Singapore? Isn’t that some little island in Asia?

Primary Mathematics is based on program of study introduced by the Ministry of Education in Singapore in 1981, a time when Singapore’s students were middling in math. Fifteen years after the adoption of its new Primary Mathematics Syllabus, Singapore students led the world in global Math achievement tests (Singapore topped international rankings first in 1995, and again in 1999, 2003 and 2007).

The Singapore math success story—from mediocre to world-class in a generation—is no secret. The curriculum provides students with a solid foundation in mathematics by focusing on visual understanding, connections, number sense, mastery, and word problems.

Concepts in Singapore math are taught in a concrete – pictorial – abstract sequence

Hands-on manipulatives or real life objects are used to demonstrate the concept, then students use and create pictorial representations. This interim visual step is typically missing from many curricula used in the U.S. It provides a transition from the words to an abstract algorithm. The goal is always to use the concrete and visual components  to get to a standard algorithm.

To gain number sense, students are taught to make connections between topics. While first graders will still work on “fact families”, Singapore math also uses a pictorial representation called a “Number Bond” to help students see the connections between addition and subtraction.

Fact Families:                  Number Bonds:

Understanding numbers and operations is critical to mathematics. Singapore materials focus on place value to provide a deep knowledge of numbers. As students work with and manipulate numbers, they work towards fluency by learning and using mental math strategies.

For example:

“If I know that 7 and 3 make 10, I could solve the problem of 47 + 8 by breaking the number 8 apart into 3 and 5. Adding the 3 to 47 gives me 50, then I can easily add on 5.”

These mental math skills show flexible thinking and provide a “check” students use when the algorithm is learned. I was in a first grade classroom last week where the teacher was talking about addition and subtraction strategies with her students. They were working with numbers like 9 + 5 and the teacher had asked the students how they got their answers:

“I counted on from 9”
“I took 5 apart to 1 and 4 and made a ten first”
“I used automaticity!”

To get to mastery, students work on focused concepts and skills. U.S. curricula are typically criticized for being “A mile wide and an inch deep”. Topics continually spiral and “It’s ok if kids don’t have their multiplication facts memorized this year, we’ll reteach them again next year.”

And next year and next year…

Not so with schools using Singapore math. In first grade, students will learn and master multiplication of twos and threes within 40. In second grade, they’ll master multiplication and division by 2,3,4,5 and 10. Each year builds on the prior foundation and extends student understanding. By the end of third grade students will have mastered all of their multiplication tables as well as multiplying and dividing by a single digit. Yep, they will even become proficient with  the dreaded “long division algorithm”.

Understanding problem solving

Another component of mastery is the ability to take what you already know and apply it in a new context. Remember being tortured in school with story problems? The heart of the Singapore curriculum is an emphasis on problem solving — and that means word problems. They are incorporated throughout the materials to provide context to each topic as it’s taught. The key to solving these begins with a bar model, or pictorial representation of the word problem. For instance:

John has 34 baseball cards and football cards in all. He has 18 baseball cards. How many football cards does he have?

John has 18 baseball cards. He has 3 times as many baseball cards as Paul does. How many baseball cards does Paul have?

2/5 of the students in a class are boys and the rest are girls. There are 35 students in the class. How many boys are in the class?

Singapore math is a great foundation for elementary math success. Working with teachers in their classrooms, I see the impact the materials have on students every day. Singapore math can help make every child in every classroom a competent and confident mathematics student.

When I go speak at a school’s Parent Night about Singapore Math, I get asked a lot of different questions. The most common one, however, is how can I help my child with his/her homework? To help out in Northern Colorado, my home base, I decided to take my Parent Boot Camps to another level. A monthly or bi-weekly level through Meetup.com. Each meetup, we’ll work through an elementary mathematical concept, review strategies, and learn a fun activity or game that supports the concept.

As we get going, I’ll post more about the meetups and the questions and issues that concern parents the most.

From the Read more about us page (where you can register to join, too):

Does your student struggle with his or her math homework? Are you arguing over who’s way of solving problems is right, yours or the teacher’s? Wouldn’t it be nice if your child thought math was FUN? It’s time for a math boot camp for parents! Each Meetup will engage parents with strategies to help their child as well as some time for home-enjoyment review. (Really, why should it be homeWORK?)

Who: Parents who want to help their elementary school students with mathematics.

Why: While math hasn’t changed much since we were in school… virtually everything else has, include teaching methods, curricula, homework, and expectations

How: Using math strategies from Singapore Math and other World-Class curricula, we’ll focus on understanding elementary math concepts. Why do we invert and multiply when dividing fractions? What are we doing when we “borrow” from a number? Why don’t kids memorize their multiplication tables anymore?

Bonus: At each Meetup, I’ll provide Math-Campers with some handpicked resources that engage students and strengthen their mastery of math facts. We’ll play games that focus on mathematics content and discuss your child’s homework assignment, so bring them along!

Overall, the CCSS are well aligned to Singapore’s Mathematics Syllabus. Policymakers can be assured that in adopting the
CCSS, they will be setting learning expectations for students that are similar to those set by Singapore in terms of rigor,
coherence and focus.

Read the full document:
Comparing the Common Core State Standards and Singapore’s Mathematics Syllabus

Achieve is comparing the Common Core State Standards to Singapore’s Math Syllabus, not the Primary mathematics curriculum materials.

I am part of a team of first grade teachers at my school. We are piloting Singapore Math this year. We are creating assessments and were looking for ready-made assessments to genuinely fit the curriculum.

One of the challenges for a classroom teacher using the U.S Edition is that there aren’t any assessments ready-made. The Standards Edition of Primary Mathematics has assessments, however they won’t correspond directly to the U.S. Edition Materials.

Keep in mind that if you write the questions, you can tailor them to the objectives of your lessons. Are you assessing for basic competency? Are you assessing for deeper understanding? Additionally, for a teacher-created assessment, you need to create an answer key. I frequently hear from teachers who tell me, “Well, I gave this test, but all I have is the answer and I’m not sure how to work the problem.”

That being said, I know teachers’ time is valuable. Many teachers use questions from the Review sections of the textbook. Other basic questions from Intensive Practice books or practice problems from the Challenging Word Problems books could be used as well.

Be sure to save any well-written assessments for future use. After one year of working with the materials, you should have a good bank of assessments for future years.

One site that offers Singapore test paper problems for free (with registration) is Old School. At the first grade they have several mid-year and end-of-term papers available. Teachers can select the questions, either short answer or multiple choice, and create an assessment.  Additionally, you can choose problems from a list of topics. Topics for first grade are:

• Addition and Subtraction
• Combination Questions (3%)
• Comparing and Ordering numbers (13%)
• Division (1%)
• General (1%)
• Graphs (5%)
• Measurements (7%)
• Multiplication (3%)
• Notation and Place values (9%)
• Shapes and Patterns (12%)

Would you be interested in a test bank of questions? How about a place for sharing some teacher-made tests?

Out in Left Field posts a Math Problem of the Week comparing different curricula that schools use. Last week’s Riddles in 2nd Grade Investigations vs. Singapore Math and the ensuing comments brought up discussion on the value of writing in the mathematics classroom.  I picked up a book entitled: Journal Writing in the Mathematics Classroom (Primary) when I was in Singapore. It is written and published by professors at Singapore’s National Institute of Education.

Among the chapters listed:

Why use Journal Writing?

Journal writing reinforces the learning and provides pupils with opportunities to engage in reflection, question their own understanding, connect the abstract and the concrete, and apply the knowledge they have acquired to solve problems.

How to Carry Out Journal Writing?

This section starts with an this powerful statement about journal writing in the classroom:

Journal writing is a complex process that requires effort and patience.

The authors further suggest that these open-ended prompts encourage pupils to write about their opinions and feelings on mathematics. They have adapted and describe three types of general writing prompts:

1. Affective or Attitudinal (How do you feel?)

• My best kept secret about math is …
• If math could be a colour (shape, sound) it would be … because

2. Mathematical content (What is it about?)

• How would you describe a …
• What patterns do you notice in …

3. Process (Explain how!)

• Find something that you learned today that is similar to something you already knew.
• You know several ways to … Which method is you favourite? Why?

There is a list of 15 of each type of general writing prompt. The bulk of the book, however, focuses on specific writing prompts that are based on mathematical topics.

Possible Negative Aspects of Journal Writing

(Lessons learned the hard way when I was teaching!)

a. The potential for the teacher to hurt pupil’s feelings.
b. The loss of instructional time to teach syllabuses
c. Tremendous increase in the marking load of the teacher.
d. Emphasis on language proficiency

Scoring Rubrics and Student Examples

This section includes examples of two types of scoring rubrics: Analytic, which allows for separate evaluation of selected factors and Holistic, which can be used when teachers want to rate student responses more generally.

A Collection of Specific Writing Prompts

Finally, there are 55 specific writing prompts differentiated by grade level and topic. Topics include: Whole Numbers, Fractions, Decimals, Percentage, Ratio, Rate, Measurement, Geometry, Statistics and Algebra. Some examples:

Topic: Whole Numbers
Level: Primary 2 – 6

Write a word problem and make a picture that goes with 4 x 3.

Topic: Decimals
Level: Primary 4 – 6

Find two decimal numbers between 0.2 and 0.3. How many decimal numbers are there between 0.2 and 0.3? Explain.

From publisher Frank Schaffer and Singapore Asian Publications comes a series of books: Singapore Math: 70 Must-Know Word Problems. From the back cover:

This book is designed to help students master word problems, which are often tricky and challenging…This book is perfect for students familiar with Singapore Math and for those who just need extra practice with word problems.

The 70 Must-Know Word Problems books are marked by levels that are considered a grade above for the U.S. market. For example, the Level 6 book claims to be appropriate for students in grade 7 and the Level 4 book is listed as appropriate for students in grade 5. If your child has been learning with Primary Mathematics and you’d like to pick up one of these workbooks, you should probably pick up the level that correlates to their current grade. Meaning, if your child is currently working their way through the Level 4 Primary Mathematics textbook, then the level 4 70 Must-Know Word Problems book will work as a supplement. The problems in the books are extra practice type problems, designed to give students similar to the ones in the Primary Mathematics textbooks. If you student is look for something harder, you might want to consider the Challenging Word Problems for Primary Mathematics series.

The 70 Must-Know series does not follow the Primary Mathematics lesson progression, the questions jump between concepts. Questions #3 and #47 in the Level 6 book, for example, are both on Volume. Question #2 & 5 cover the four operations with money  and Question #4 is on percentages, so you may need to verify that a topic has been covered in class if your planning on using the book at home. The books have little instruction, however they do include a four page Introduction to Singapore Math (be forewarned, it’s written in six-point font!). Each problem has its own full page for work and there are fully worked solutions included in the answer key.

Here’s Question #70 from the Level 4 book. It’s listed in the table of contents under “Addition, Subtraction, Multiplication and Division of Whole Numbers”.

At a movie, 1/4 of the people in the theater were men, 5/8 were women and the rest were children. If there were 100 more women than children, what was the total number of people in the theater?

And Question # 48 from Level 6:

A box contained some red, blue, and green markers. For every 5 red markers, there were 2 blue markers. For every 3 blue markers, there were 5 green markers.

(a) Find the ratio of red markers to blue markers to green markers.

(b) When 6 red markers were removed from the box, 3/7 of the remaining markers were red markers. How many markers were left in the box?

Singapore Math: 70 Must-Know Word Problems was$12.99 at my local Barnes and Noble and Borders stores. The easy accessibility of these books make them great for parents looking for additional problems for their students. Teachers looking to use these materials in class need to keep on mind that these books may already be in their students’ homes!

Google Books is offering free previews of books in this series:

• Singapore Math 70 Must-Know Word Problems, Level 1
• Singapore Math 70 Must Know Word Problems , Level 2
• Singapore Math 70 Must-Know Word Problems, Level 3
• Singapore Math 70 Must-Know Word Problems, Level 5

On the (SingaporeMath.com) site there are some supplemental books I don’t recognize, but I can’t tell what they’re for–if they’re new-new or replacing-CWP-new

Math Works?
i-Excel?
Math Express?
Brain Maths?

I wrote here about the MathExpress: Speed Maths Strategies.

i-Excel Heuristic and Model Approach
Author: Li Fanglan
Published by Fan-Learning
Levels 1-6 available in the U.S. from Singaporemath.com

The i-Excel: Heuristic and Model Approach is a completely different type of workbook than the MathExpress series.  i-Excel books focus on problem solving. Can you deduce the meaning of “heuristics” based on the following description from the introduction?

At primary levels, Model Approach has been proven to be the most versatile and effective method to help pupils solve many difficult problems.Heuristic Approach, on the other hand, helps them handle the higher level problem solving by unconventional means.

Heuristic Approach

The first part of each book includes eight different Heuristic Approaches and then focuses on different Model Approaches based on topics taught at each level. Heuristic Approaches in Level 2 include the following:

1. Guess and Check
2. Act it Out
3. Draw a Diagram I
4. Make a List
5. Look for a Pattern I
6. Draw a Diagram II
7. Draw a Diagram III
8. Look for a Pattern II

Upper level Heuristic Approaches include: Simplify the Problem, Work Backwards, Make a Supposition, Solve Part of the Problem, Use Before-After Concept and Restate the Problem in Another Way. Each Heuristic Approach has a worked example, four or five practice problems and one more challenging problem to solve. (Some levels put these challenging problems into a separate unit.)
Here’s the challenge problem from Level 2 – Draw a Diagram II:

A pizza was cut into halves.
Jolene took one half and ate 2/3 of it.
a) What fraction of the pizza did she eat?
b) What fraction of the pizza was left?

Completed guided examples are included in the Answer Key at the end of this section. Make a note, however,  not all questions have worked solutions.

Model Approach

The second portion of the books works with the Model Approach as it applies to mathematical topics.

• Levels 1-3 focus on applying the Model Approach with the four operations. They also have yellow, pink and blue rectangular stickers in the back of the book for students to use with the model drawing problems. These are important for students who are learning or struggling with drawing proportional bar models.
• Level 4 has two parts consisting of 28 units on the Model Approach: Whole Numbers and Fractions. There is also a Part IV: Non-Routine Problems – Challenge Yourself 4.
• Level 5 has sections on Whole Numbers, Fractions, Ratio, Decimals, and Percentage. Part VII includes 3 assessments.
• Level 6 parts include ratio and Proportion, Percentage and Speed. The final section is entitled “Examination Practice”. Some of these problems are double starred for extra challenge.

Here’s an example of a ** problem from that unit:

Grace had a total of 120 red and blue pens in the ratio of 3:5. After she gave away an equal number of each type of pens, the number of red and blue pens left was in the ratio 3:8. How many pens did she give away altogether?

Good news if you’re scratching your head right now. All problems in Level 6 have detailed solutions worked in the answer key.

In the classroom

Most who have used Primary Mathematics would agree; the focus is on the bar model as the main problem solving strategy. The i-Excel series brings explicit instruction in additional problem solving strategies into the classroom. This is one of my favorite supplemental books to use with students. The challenge and variation makes it a favorite of students as well. I have incorporated the heuristics into a self-directed activity and have used the challenge problems for a “Problem of the Week”.

The Level 3 book includes a unit entitled “Act it Out” that became a great independent activity/group center. The example given is to use 10 coins to form the figure below. Moving only one coin at a time, what is the least number of moves to turn the shape upside down?

After working this example as a whole group activity, I could now have chips (coins) and new problems available in the classroom: as part of centers, as a substitute lesson plan, or for students to use as a quiet desk activity.

Have you used the i-Excel or Brain Maths series? Share your experiences in the comments below. I’d love to know how the books have worked in a classroom or in your home.

On the (SingaporeMath.com) site there are some supplemental books I don’t recognize, but I can’t tell what they’re for–if they’re new-new or replacing-CWP-new

Math Works?
i-Excel?
Math Express?
Brain Maths?

First, these aren’t replacing CWP (Challenging Word Problems series), but they are fun books that are great supplements for homeschooling families or classrooms.

MathExpress: Speed Maths Strategies
Author: Li Fanglan
Published by Fan-Learning
Levels 1-6 available in the U.S. from Singaporemath.com

I recommend MathExpress if you are interested in becoming faster and more fluent with mental mathematics. Book levels 1 – 3 focus on basic mental math strategies with the four operations. Beyond that level, if you’re using them with students, some of these approaches can start making math look like a series of algorithms to memorize or tricks. It’s important that students understand the reason why these “short-cuts” (as they are referred to in the books), make mathematical sense.

An example from Level 1- Express Strategy 13:

Can you get the answer in 10 seconds?
26 + 49 = ?
58 + 37  = ?

There is a visual & written solution for each problem on the next two pages. Here’s the first written solution:

1 and 49 make 50.
Rewrite 26 as 25 + 1.
Add 1 to 49 to get 50 before adding 25.

An example from Level 2 – Express Strategy 9:

Can you get the answer in 10 seconds?
342 – 190 = ?
237 – 172 = ?

Again,  there is a visual and written solution page for each problem. Here’s the first:

190 is 10 less than 200. Subtract 200 from 342 before adding 10.

After an explanation of the strategy, there is a page of practice, a page with two word problems and a Speed & Accuracy Test.

There are six volumes in the series:

Levels 1 and 2 include addition and subtraction.
Level 3 adds in multiplication and division.
Level 4 includes all four operations and decimals.
Level 5 has fractions and decimals.
Level 6 has strategies to check answers, percentages & advanced problem solving. Here are two 10-second problems from Level 6 and the first solution:

125 x 25 x 32
1/2 x 50 x 28 x 11

4 and 8 are factors of 32.
24 x 4 = 100 and 125 x 8 = 1000.
Multiply 25 by 4 and 125 x 8 before multiplying the two products.

Before working the solution to the second problem, the book provides this word problem:

A rectangular tank measuring 50 cm by 28 cm by 11 cm is half filled with water. Find the volume of water in the tank.

I highly recommend the upper level books for adults looking to improve their mental math abilities. I should also mention that the books have two diagnostic assessments at the back of the book, along with an Answer Key and Detailed Solutions. (You’ll find the solution to the 6th grade level problem above on 69.)

In the classroom

My students have so much fun working on these strategies. In a classroom, I’ve used these books with students AFTER concepts have been mastered to help students become faster with their mental computation. As an example, in a 3rd grade classroom, I would focus on the strategies in the Level 2 book for the first half of the year, then, depending on the students’ understanding of multiplication and division, I’d introduce some of the strategies from Level 3. These can conclude in wonderful mathematical conversations. Here’s an Express Strategy from Level 3 that should lead to an interesting discussion:

Can you get the answer in 10 seconds?
26 x 5 = ?
148 x 5 = ?

And the strategy:

2 fives = 10
Multiply 26 by 10 instead, then half the product.
26 x 5 = 26 x 10 ÷ 2

= 260 ÷ 2
= 130

I’ve used the i-Excel and Brain Maths series. I’ll post reviews on those also. If you’ve used these materials, please share how in the comments below. I’d love to know how they worked in a classroom or in your home.

Old School aims to be the premier resource for FREE educational material from Singapore, especially Singapore Math and Singapore Science. We have a large database of test questions and assessments and are always adding new content.

Currently, they offer test papers as well as the ability to take tests online in English, Mathematics and Science. Registering on the site allows you to track your progress on the online tests (usually the “A” part of each). Best of all, you can search the questions by topic and select questions with either a multiple choice or free-response format.

As an example, the Primary Four 2009 Mathematics page offers:

• 3  Continual Assessment 1 tests
• 4  Mid-Year Examination tests
• 3  Continual Assessment 2 tests
• 4 End Year Examination tests

That include the following topics (with the percentage of questions):

• Angles (7%)
• Area and Perimeter (8%)
• Decimals (4%)
• Factors and Multiples (6%)
• Four Operations (22%)
• Fractions (18%)
• Geometry (2%)
• Graphs (2%)
• Measurement (7%)
• Non-Standard Questions (6%)
• Perpendicular and Parallel Lines (4%)

The majority of mathematics papers are a 2 out of 4 on the site’s scale of difficulty. There were some assessments with a difficulty rating of 1.

Here’s a problem from the 2009 Primary Four End Year Assessment for your enjoyment:

Fill in the missing number.

108 x 99 = 110 x 99 + 10 x 99 – ( ? ) x 99

1. 8
2. 2
3. 12
4. 228

The materials are all from the third grade level:

1. Primary Mathematics U.S. Edition (2003)  from SingaporeMath.com
2. Primary Mathematics Standards Edition (2008)  from SingaporeMath.com
3. My Pals Are Here Maths (2007) obtained in Singapore from Marshall Cavendish Education
4. Shaping Maths (2007) obtained in Singapore from Marshall Cavendish Education

Following are the practice pages from each workbook that correspond with the lesson on addition within 10,000 that introduces regrouping in the hundreds. As before, each thumbnail links to a full-sized file.

Once again, there are minimal differences between the U.S. and Standards editions of Primary Mathematics.  Problem #1 changes pictures from towels hanging on a clothesline to boats. Problem #2 has one small change. The equation for  letter B changes from  4107 + 5 to 4105 + 5. Finally, on problem #4, “Weihua” becomes “Will”

U.S. Edition Workbook 3A:

Standards Edition Workbook 3A:

The My Pals Are Here Workbook is perforated and 3 hole punched. Perforated pages would be a great change to make to the Primary Mathematics workbooks! Neither My Pals Are Here nor Shaping Maths have any word problems tied to this practice lesson, in fact, there are very few word problems in the books at all.

My Pals Are Here Workbook 3A Part 1:

Shaping Maths Activity Book 3A part 1:

Were you expecting less practice in the materials from Singapore?

Parts in the series:

Part 1 – Teacher’s Guides
Part 2 – Textbooks
Part 3 – Workbooks
Part 4 – Supplemental materials