An Introduction to Singapore Maths

During your child’s year in first grade, you will have many opportunities to learn more about the Singapore Math curriculum.  Here is a brief introduction:

• Singapore Maths revolves around several key number sense strategies: (1) building number sense through part‐whole thinking, (2) understanding place value, and (3) breaking numbers into decomposed parts or friendlier numbers, ones that are easier to work with in the four operations of addition, subtraction, multiplication and division.

• Singapore Maths does something dramatically different when it comes to word problems. It relies on model drawing, which uses units to visually represent a word problem. Students learn to visualize what a word problem is saying so they  can understand the meaning and thus how to solve the problem.

• Mental math teaches students to calculate in their heads without using paper and pencil. Your child will still need to commit some facts to memory, but mental math will teach him or her to do calculations using proven strategies that don’t require pencil and paper.

• Strategies taught in Singapore are layered upon one another. One strategy is the foundation for another one.

• Singapore Math teaches students to understand math in stages, beginning with concrete (using manipulatives such as counters, number disks, cubes, and so on), then moving to pictorial (solving problems where pictures are involved), and finally working in the abstract (where numbers represent symbolic values).

Through the process, students learn numerous strategies to work with numbers and build conceptual understanding. With time and practice, they eventually master the traditional methods and algorithms.

While we differentiate instruction to meet individual student needs, the chart on the opposite page is a road map to show progression of computational strategies across the grade levels:

Kindergarten	Making bonds to build numbers 1-10. Using the place value mat.
First grade	Making bonds to build numbers 11-20. Branching bigger numbers into smaller ones to make them more manageable. Using place value mats to decompose, add, and subtract numbers. Adding horizontally from left to right (instead of adding vertically).
Second grade	Using place value mats for more complex addition and subtraction. Branching with even larger numbers. Adding vertical addition to the mix. Using model drawing.
Third grade	Using place value mats with multiplication and division problems. Branching with larger numbers. Using model drawing. Using the distributive property to decompose numbers.
Fourth grade	Using the area model for multiplication and using partial quotient division. Using model drawing for increasingly complex problems.
Fifth grade	Using model drawing proficiently.

Examples of First Grade Strategies

Number Bonds: Part-Whole Thinking The beginning of number sense is viewing each digit as a part of a whole. This is very similar to fact families, where a number has specific “relatives” in its family. Let’s take the number 6 as an example. 6 is 6 plus 0, 5 plus 1, 4 plus 2, and 3 plus 3. This understanding becomes very important when students are challenged to use operations with the number 6. After students learn digits 1–9, they master what combinations make 10. 10 is an anchor number in Singapore as we use a Base 10 system. In K and grade 1, students will spend a significant amount of time learning their bonds through 10. 

  

Branching: Students spend time learning how to break numbers into place value groupings on the place value board. This is called decomposing numbers or using expanded notation. After students practice breaking numbers apart into place value groupings, we teach them to add and subtract by place value. This is branching. The goal with branching is for students to break numbers into place value groupings and then do the operation with those place value groups. For example, 23 + 42 would be branched into tens and ones. Then students will add and then add the groupings together.

23 + 42 =(20 + 3) + ( 40 + 2)

20 + 40 = 60 + 3 + 2 = 5

60 + 5 = 65

The goal for branching is for students to eventually be able to look at the problem and work it out mentally.

Place Value: Singapore Maths is a Base 10 system. A number’s place value is determined from right to left, starting with the ones and moving through the tens, hundreds, thousands, ten thousands, one hundred thousands, to a million and beyond. In class, we use tools such as place value boards with disks and cards to help us organize, visualize, and understand value of numbers and how they relate to one another.

Algorithms An algorithm is a systematic, step‐by step procedure to solve a problem using a mathematical operation. For example, with subtraction, we have learned to line our numbers up vertically so that the digits are in the correct place value columns. We have traditionally learned to subtract the digits moving from right to left, using regrouping or borrowing, in order to get the correct answer to the problem. In Singapore, traditional algorithms are taught and mastered with the help of the place value mat. However, we also teach alternative algorithms or strategies to solving equations often before we teach the traditional ones. This helps us build and reinforce our understanding of number sense and place value. This also allows students to use a strategy that they are competent at using for any problem. Rather than having one strategy, they may have several to choose from, and they can use the one that’s most intuitive for them.

Model Drawing: Model Drawing is the key strategy we use to solve word problems.

Read the problem to get a sense of what is asking.

Decide who and what the problem is about.

Draw units for each who and what.

Reread the problem and adjust our units to match the word problem.

Decide what the question is asking of us and place a ? in place.

Work our computation.

Write our answer in a complete sentence

Mental Math: Mental math is one of the cornerstones of Singapore Math as its emphasis is on helping students to calculate mathematically in their heads, thus developing number sense and place value. It encourages flexibility and speed when working with numbers. We practice mental math strategies and do lots of fun activities that support the skill. Please come to class and participate with us one day! 

Singapore Maths Compared of Purposeful Design

INTRODUCTION:

SOUTH AFRICA:

The World Economic Forum’s annual report on financial development, released in October 2012, placed South Africa last in a ranking of 62 countries in the quality of mathematics and science education. The forum’s report pointed to a high correlation between human capital and the degree of financial development in countries.

The 2011 census results, released in 2012, showed that a decade ago 2.7% of men and 2% of women who had tertiary qualifications in South Africa had qualifications in the fields of natural, physical and mathematical sciences.

By 2011, this had declined to 2% of men and 1.8% of women On 17 April 2013, the Mail and Guardian reports: South Africa has ranked its maths and science education has second last in the world, ahead of Yemen, according to a World Economic Forum report. Forty-three percent of South African grade five learners failed to reach the lowest international benchmark, in contrast to 5% of grade four learners internationally.

This means that they have not yet mastered the basic reading skills required to access and retrieve information for reading comprehension purposes," the Pirls (Progress in International Reading Literacy Study) report stated.

It further revealed that about 90% of the grade four learners tested in English or Afrikaans attained the lowest international benchmark, while between 24% and 57% of children writing in all nine official languages did not achieve it.

Learners tested in Sepedi and Tshivenda achieved the lowest results, according to the study.
In December 2012 the Department of Education has released the Annual National Assessment (ANA) results, which provides a bleak picture of educational performance. The performance in mathematics is however extremely worrying. The results show that the largest number of South African students have been assessed to score below 30% in Grades 3, 6 and 9. When a student scores between 0-29%, this is described as “not achieved”. The scoring used by the Department of Basic Education is as follows:

Rating Percentage Description
Level 1 0-29 Not achieved
Level 2 30-39 Elementary achievement
Level 3 40-49 Moderate achievement
Level 4 50-59 Adequate achievement
Level 5 60-69 Substantial achievement
Level 6 70-79 Meritorious achievement
Level 7 80-100 Outstanding achievement

The graphic below summarises the data, showing that 9 out 10 South African students in Grade 9 do not even score above 30% in mathematics.
                                         
The data paints a worrying state on mathematics education in South Africa. The performance across Grades 3, 6 and 9 indicate that performance levels drop as students move to higher grades.

SINGAPORE:

Singaporean students ranked first in the world in mathematics on the Trends in International
Mathematics and Science Study-2003 (TIMS); U.S. students ranked 16th out of 46 participating nations at grade 8 (Mullis, et al., 2004). Scores for U.S. students were among the lowest of all industrialized countries. Because it is unreasonable to assume that Singaporean students have mathematical abilities inherently superior to those of U.S. students, there must be something about the system that Singapore has developed to teach mathematics that is better than the system we use in the United States.

Singapore’s superior performance on the Trends in International Mathematics and Science Studies (TIMSS) in 1995, 1999, and 2003 affirms the high quality of Mathematics education in Singapore (Ministry of Education Singapore, December 16, 2004).

Analysis of these evidentiary streams finds Singaporean students more successful in mathematics than their U.S. counterparts because Singapore has a world-class mathematics system with quality components aligned to produce students who learn mathematics to mastery. These components include Singapore’s highly logical national mathematics framework, mathematically rich problem-based textbooks, challenging mathematics assessments, and highly qualified mathematics teachers whose pedagogy centers on teaching to mastery. Singapore also provides its mathematically slower students with an alternative framework and special assistance from an expert teacher.

Singapore’s mathematics framework defines expectations about what students should know and be able to do in mathematics. Singapore’s well-defined syllabus describes mathematical topics and outcomes grade by grade within broad mathematical strands. Singapore gives its teachers much of the credit for its education success. Singapore’s teachers “lie at the heart of all we do in education” (Ministry of Education, Singapore, 2001c).

UNITED STATES OF AMERICA:

The U.S. mathematics system does not have similar features. It lacks a centrally identified core of mathematical content that provides a focus for the rest of the system. Its traditional textbooks emphasize definitions and formulas, not mathematical understanding; its assessments are not especially challenging; and too many U.S. teachers lack sound mathematics preparation. As a result, the United States produces students who have learned only to mechanically apply mathematical procedures to solve routine problems and who are, therefore, not mathematically competitive with students in most other industrialized countries.

However the U.S. mathematics system has some features that are an improvement on Singapore’s system, notably an emphasis on 21st century thinking skills, such as reasoning and communications, and a focus on applied mathematics. However, if U.S. students are to become successful in these areas, they must begin with a strong foundation in core mathematics concepts and skills, which, by international standards, they presently lack.

COMPARISON BETWEEN PURPOSEFUL DESIGN MATHEMATICS and SINGAPORE MATHS

ACSI wanted to ensure that the curriculum we offer our schools via Purposeful Design is indeed the best in comparison with world standards. It was important for us to ensure that we remain true to our calling in offering an alternative that is Bibically sound and distinctively Christian as wel as a rigiourously intellectual and  academic.

We have drawn the following conclusions:
• Purposeful Design’s Mathematics Curriculum  meets or exceeds all of the national standards and has documented its close alignment to the Common Core Standards in the USA.

Both series are strong in problem solving.
• Singapore Math uses a specific square model for problem solving
• Purposeful Design also provides a specific problem solving guide. As a new concept is taught students use that concept in solving problems.

• Feedback from schools moving away from Singapore Math is that parents sometimes find it difficult to help their children with homework because of the teaching methods employed.
• Feedback from upper level math teachers after Purposeful Design  has been implemented at lower grade levels is that they notice a change in the foundational learning of students using Purposeful Design  mathematics, especially in the area of fractions and abstract concepts

• Singapore math assumes that once major concepts are taught, they do not need to be retaught.
• Purposeful Design provides a regular review of all concepts.

Advantages of Purposeful Design Mathematics
• Purposeful Design provides solid understanding of concepts through the use of manipulatives and modeling.
• Differentiated instruction is integrated into the “Directed Instruction” component of the lesson, addressing the various learning styles of students in the classroom.
• Basic facts and operations are taught while developing concepts through the use of manipulatives, investigation, and hands-on activities.
• Algebra and geometry strands are woven through the entire series, beginning in kindergarten.
• Tools for enrichment, recovery and practice are provided.
• Emphasis on math vocabulary through the use of math journals at each grade level.
• Designed to be teacher, not textbook-driven, Purposeful Design  provides the necessary tools for teachers to be successful.
• Since the series is concept-based, the majority of the lesson is often taught through manipulatives and modeling before students open their texts. Proper use of manipulatives and modeling are provided for in the Teacher Edition.

Contributions made by:
• Marikita Evangelista: Retired Professor: De La Salle University, Manila Phillipines
• Sheilamarie Uy Eiaw: Human Resource Department Head, MGC New Life Christian Academy, Metro Manila, Philippines.
• Cindi Banse: Math and Science Support Specialist, ACSI Colorado Springs

Compiled by:
• Anna-Marie Russell: ACSI Director South Africa