*The Daily Riff is featuring a four-part series*

**Singapore Math Demystified!**by Bill Jackson, Math Helping Teacher, Scarsdale, NY Public Schools, one of the highest performing districts in the country, due to popular interest in this topic from parents and schools. If you missed Part 1,__Why I Became Interested in Singapore Math__, check it out HERE.*For non-math types, don't let the pentagon below "throw you" - just skip it - Bill explains Singapore so clearly and simply for parents AND for educators. Math-phobia is one of the more challenging issues in this country's educational make-up and what better way to learn than from countries such as Singapore, which ranks top internationally. By the way, "pedagogy" is synonymous with "teaching methods" so nothing mysterious about that! Include the word in one of your next parent-teacher conferences and your teacher may find the conversation more intriguing and certainly more challenging. - Ed. Note*

**Philosophy and Pedagogical Approach of Singapore Math**

**Part 2**

by Bill Jackson

In 1989, the National Council of Teachers of Mathematics (NCTM) published the Principles and Standards, which advocated a problem-solving approach for the teaching and learning of mathematics. Thoughtful application of this approach, however is still relatively uncommon in U.S. classrooms. Analysis of the TIMSS videotape study revealed that Japanese math lessons embodied the spirit of the NCTM Standards more than American lessons. How could this be, given that NCTM is a U.S. organization?

**It turns out that while American teachers were still trying to make sense of the standards, Japanese teachers were studying how to make them come alive in their classrooms through lesson study.**

Another nation that took the NCTM Standards and other research on problem-based approaches seriously is Singapore. After its independence in 1965, Singapore realized

that without any natural resources it would have to rely on human capital for success,

so they embarked on an effort to develop a highly educated citizenry. Various education

reforms were initiated and in 1980 the Curriculum Development Institute was established,

which developed the Primary Mathematics program. This program was based on the

concrete, pictorial, abstract approach. This approach, founded on the work of renowned cognitive American psychologist Jerome Bruner, encourages mathematical problem solving, thinking and communication.

The fact that problem solving is the central idea in Singapore math can be seen in the

pentagon from Singapore's Mathematics Framework below.

The Framework states, "

*. It involves the acquisition and application of mathematics concepts and skills in a wide range of situations, including non-routine, open-ended and real-world problems. The development of mathematical problem solving ability is dependent on five inter-related components, namely, Concepts, Skills, Processes, Attitudes and Metacognition."*

**Mathematical problem solving is central to mathematics learning**Problem solving and mathematical thinking are two big ideas behind Singapore math. To understand this better, let's look at an example that many American elementary students struggle with, long division.

**or if they do, usually only the advanced students have the opportunity to tackle them while struggling students continue to practice procedures. The third grade Primary Mathematics textbook, however, introduces long division with a word problem. The description below is one way how the concept of long division might be introduced in**

*As noted in part one of this blog, word problems are often the*

last thing on the page in U.S. mathematics textbooks. Often times teachers never even

get to these problemslast thing on the page in U.S. mathematics textbooks. Often times teachers never even

get to these problems

Singapore math.

After a brief warm up with multiplication and division flash cards, the teacher introduces a problem by saying, "Our friend Meihua has some toy soldiers. She wants to put them

equally in some tents." (Note that no numbers are mentioned and there is no question asked yet.) The teacher then asks the students to try to imagine the situation and discuss what it means to put the soldiers in tents equally. Students share examples such as, "If she has 15 soldiers and 3 tents, she could put 5 soldiers in each tent," and "If there were 10 soldiers

and 5 tents she could put 2 soldiers in each tent."

Next, the teacher gives the students 14 counters (chips) to represent the soldiers and 4 cups

to represent the tents. Then she poses the problem, "Meihua has 14 toy soldiers. She

puts the toy soldiers equally into 4 tents. How many soldiers are there in each tent?

How many soldiers are left?"

Students begin solving the problem by dividing the counters among the cups as well as any other methods they come up with (e.g. using the multiplication table of 4). Then,

*several students share their solution methods with the class, the methods as well as errors are discussed and evaluated, and important ideas are highlighted.*The teacher then uses

these ideas to introduce the division algorithm for 14÷4, relating each step to the soldiers

and the tents. Lastly, vocabulary words such as dividend, divisor and quotient are introduced and related to the problem.

Solving and discussing this "anchor problem" has taken a little more than half of the one-hour math period. For the rest of the class, students solve and discuss a few carefully selected problems from the textbook, gradually moving from simple to more complex numbers, and using pictures from the textbook to help them form a mental picture of the process. Over

the course of the unit, students will learn to calculate long division problems using only

numbers but they will have frequent opportunities to move through this cycle of concrete manipulation to pictorial representation to abstract calculation.

*Since there are fewer topics*

in the Singapore math textbook, the students have sufficient time to study the concept in depth to understand and master it.in the Singapore math textbook, the students have sufficient time to study the concept in depth to understand and master it.

I would like to point out a few reasons why this approach is helpful.

1) The problem situation provides a familiar context for students to think about an abstract concept.

Although 14÷4 might seem easy to adults, for a third grader, especially one who is already struggling in mathematics, it may seem quite abstract. The use of a familiar situation makes it easier to understand. Every child can think about dividing toy soldiers equally into tents. Since no numbers are mentioned initially, no child is lost. Every child can enter the lesson successfully at some level.

The problem setting also makes mathematical terms like divisor, quotient and remainder

easier to understand.All too often,The toy soldiers become the dividend, the tents the divisor, the number of soldiers in each tent the quotient, and the soldiers who don't have a tent the remainder. When used in this way, instead creating difficulty,abstract mathematical concepts and vocabulary are presented in a void.But the problem situation gives the terms meaning.the word problem makes learning more concrete by presenting abstract ideasAs a result, students' attitudes towards problem solving and mathematics in general improve.in a familiar context.

2) By sharing and discussing their solution methods, students develop metacognitive

processes (ed. note: metacognition - awareness & understanding about one's own

thinking).

Sharing and discussing their methods requires students to think about their metacognition, as well as the thinking of their classmates. When students share their own solution methods they are required to make their thinking clear and explicit so

their classmates will understand them. When they have to listen to their classmates' methods and restate their friends' thinking in their own words they learn how to listen

to and learn from each other. The pentagon from the Singapore Mathematics Framework shows how important reasoning, communication, thinking and metacognition are for students to become good mathematical problem solvers.

3) By spending time on concrete manipulation and pictorial representation, students are able

to internalize and visualize mathematical concepts.

The concept of division can be very abstract to children. Thinking about division using counters and cups is much easier than thinking about it with abstract numbers. By manipulating concrete objects, students internalize the division process.U.S. math programs often also begin with concrete activities but students usually go from using concrete objects right to abstract calculation.Singapore math textbooks include an intermediary pictorial stage.By looking at pictures of concrete objects being divided equally, students form a mental image of what long division looks like. When they finally get to abstract calculation, they have already internalized and visualized the process.

*The above example shows a little of what Singapore math is. But I would also like to discuss what it is not, in order to dispel a couple of common misconceptions:*

1) Singapore math is

**not**the way we learned math as kids (that is, unless you went to school

in a developed East Asian nation).

In Singapore math, traditional algorithms are learned but conceptual understanding is taught before procedural fluency. For example, the long division algorithm taught in Singapore math textbooks is the same one we traditionally learn in the U.S., but it is presented and explained quite differently than the way most of us learned it (see Power Point presentation below). In Singapore math, students not only learn how to do an algorithm but also why and how every step of the algorithm works.

2) Singapore math is not "drill and kill," facts memorization, or rote learning of procedures.

In Singapore math, students learn their addition, subtraction and multiplication facts and eventually memorize them. But they also learn the structure and patterns behind the facts so if they forget them, they are able to reconstruct them in their minds. From early grades, children learn how to compose and decompose numbers and manipulate them in useful waysin order to calculate mentally.

Singapore math is about students solving problems, thinking deeply, sharing their ideas, and learning from one another. Conceptual, procedural and factual understanding is developed through problem solving and carefully structured practice and as a result, students learn how to think deeply and appreciate mathematics.

We cannot to go into sufficient detail in this blog about how mathematical concepts are presented in Singapore math so I am attaching a SingMathPPT.pdf presentation that will give you an idea of how basic algorithms and mental calculation are taught. (large file so download may take a couple minutes). In future posts, I will discuss the use of bar models for problem solving, and tips for successful implementation for schools that are interested in using Singapore math.

Until next time,

Bill Jackson

Math Helping Teacher

Scarsdale Public Schools

Part 1: Singapore Math: Can It Solve Our Country's Math-phobia?

Part 3: "The Famous Bar Modeling: Is Singapore Math The Most Visual Math?"

Part 4: Bringing Singapore Math to Your School + Tips For Teachers

Also check out Bill Jackson's recent travel Journal: "Singapore: Five Surprises In Education"

Related: Bill Jackson's Travel Journal: "What American Teachers Can Learn From Japan"

Thanks Bill, not only do our kids feel more comfortable with math, but we do as well. I would like to see our lesson study expand into all areas. You are the best and I hope these articles don't make you so famous that you leave us. - Kate

My problem is not how to present the problem. If I started with step 1, and asked my students to share ideas on how to divide soldiers into tents. They will say,

"Oh this is gay!", and some will fake staying on task, and others will openly say this is boring, i don't want to do it.

My problem is the attitude, side of the pentagon!

Please do not try to give write me a prescription for building attitude. The problem is the fact that they have counted the teeth of the horse of the American education system. They know that they will have to be passed ,no matter what. Why even bother, they claim. They are not stupid!

You ask this of any teacher at my school(the non magnet teachers), they will tell you the same thing. You need to have the "Attitude" on the 8 sides on an OCTAGON, and come up with a whole new system of education in California.

Thank you for your comment. I do not believe there is a "prescription" for building attitude, so I will not try to provide one. It is complex, of course, but this is the art of teaching (and presumably why we entered this profession)--to engage students, hold their interest, and promote critical thinking and real learning, not as a way to make the grade but as a way to fully develop our human potential. I have done this with inner city, high needs students as well as with students from wealthier backgrounds so I know that although it is not easy it can be done. Once students understand that it is not about getting a grade or passing, they can relax and learn. The focus should be on mathematical thinking and all students can think mathematically.

Thank you for your comment. I do not believe there is a "prescription" for building attitude, so I will not try to provide one. It is complex, of course, but this is the art of teaching (and presumably why we entered this profession)--to engage students, hold their interest, and promote critical thinking and real learning, not as a way to make the grade but as a way to fully develop our human potential. I have done this with inner city, high needs students as well as with students from wealthier backgrounds so I know that although it is not easy it can be done. Once students understand that it is not about getting a grade or passing they can relax and learn. The focus should be on mathematical thinking and all students can think mathematically.

I remember a good friend of mind who didn't work out the mechanical problems in his calculus book before the exam, but he knew the theories and how to develop answers. He went in, applied the theories and came out with an "A". Part of developing a knowledge of theory is visualization - perhaps good 50% of it. Project-oriented learning also does this. Whatever the case, rote memory and mechanics are the worst ways of learning. Not only in elementary math but in symbolic logic, the same ineffective methods of memorizing formulas exist. Never did I have students memorize but they learned how to develop inference rules on their own - thousands of them through the basic theories and visuals, instead of the 18 or so that student memorize out of the average logic text. All my exams were open book. Students usually anticipated the next part of the course, as they were learning the theories so well and could apply them. So even though I didn't use Singapore math per se, I know the methods work from experience. I say bring the whole system to every U.S. school. We should have done this yesterday.

Hi Jeremy,

Thank you for this insightful comment. I recently returned from Singapore where I heard master teacher Ban Har Yeap say that the job of the professional developer in Singapore is to change teachers' beliefs, specifically the common misconception that mathematics is about learning procedures, rules and drill. Mathematics is a habit of mind, a way of thinking, a way of enriching our souls.

Bill

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