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, "Mathematical problem solving is central to mathematics learning. 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."

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. 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 problems 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
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.

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, abstract mathematical concepts and vocabulary are presented in a void. But the problem situation gives the terms meaning. 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, the word problem makes learning more concrete by presenting abstract ideas in a familiar context. As a result, students' attitudes towards problem solving and mathematics in general improve.

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 ways in 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

THE KIDDIE TOES MONTESSORI ADVANTAGE

DID YOU KNOW THAT one year of mental growth from age three to four is equivalent to three years of growth between ages 9 to 12? Don't let your kids lose out on the best years of their lives.

Here at KIDDIE TOES MONTESSORI SCHOOL we specializes in providing a "Prepared Environment" that spurs your child to learn during his most receptive stage from 3 to 6 years - a period when his mind leaps to growth. This environment is modified to suit his later developmental needs between 7-12 years, 13-16 and 17-20 years of age.

One of the factor missing in traditional schools is the Montessori "Prepared Environment" which readily conditions the child to work and concentrate. Bad habits of disorderliness, selfishness or dependence will be replaced by obedience and friendliness. Thus, the Montessori discipline is acquired through work not play.

Friendships - An Important Part of Childhood

Friendship is one of life’s greatest gifts. I was reminded of this the other day when two colleagues disclosed that they had been friends since kindergarten and had shared the ups and downs of school years, teenage years, marriage, children and careers. Considering they are both baby boomers, that is some test of time. By today’s standards where everything is disposable, dispensable, recyclable and replaceable, it was quietly comforting to hear of something that had remained constant. When we finally roll up to the pearly gates and reflect on our lives, I daresay it won’t be the material things we’ll value the most, but rather the people we’ve held near and dear.

Our children are growing up in a very different world to previous generations. They can expect to change house, move schools, travel overseas, meet people from many cultures and develop cyberspace friendships all before they get the ‘key to the door.’ Previous generations had smaller lives, moved in tighter, more limited circles and befriended people primarily from their immediate neighbourhood or workplace.

The social and sporting lives of children are enough to leave the average adult’s head spinning. Friendships originate from a seemingly unlimited supply of activities: music lessons, dance practice, swimming training, sporting interests, after school care and last but not least - school.

On face value you’d think friendships would be easier to form and maintain than ever. But this isn’t always the case and chances are kids today will at some stage find themselves floundering with friendships, battling with bullies or rattled by relationships.

So how can we help our kids nurture lasting friendships?

Being a good role model is always a great place to start. Demonstrating that friends are important, showing care and consideration for the feelings of others and a willingness to compromise when differences arise, is all good stuff. Discussing ways to be a good friend in order to have a good friend, gives children sound grounding.

Encourage them to have a variety of friends. The old adage, ‘don’t put all your eggs in one basket’ makes sense, especially today, when children can move schools, states or countries before leaving primary school. Losing your best friend can become monumentally challenging and devastating if they are your only friend. Encouraging your kids to invite several children from a variety of sources into their life, is good insurance. Being watchful for others who endeavour to dominate your child’s time at the exclusion of others is sound practice. Nurturing that special bond that exists between ‘best friends’ comes naturally when the connection exists, but developing a wider social network simultaneously is equally important.

Sitting back and letting your kids fight their own battles within reason adds to their education on relationships. Falling out with friends is always tricky and with children it is doubly difficult, because you have parents to deal with as well. Unless the issue is of considerable magnitude, on most occasions, common sense and time-out can sort out most tiffs. Kids are resilient and forgive and forget at the speed of light, especially when a new game, toy or activity to explore together is on the horizon. Learning how to settle differences, negotiate with mates and compromise is all par for the course for future friendships so they might as well learn it young.

The flipside of friendship is bullying. Unfortunately it is on the rise in schools, not only in this country but globally. Bullying was once thought of as physical abuse carried out in a sly, secret way. Research has shown that emotional bullying is equally damaging. It manifests itself in many ways: teasing, name- calling, threatening, staring and excluding. The latter often inflicts the greatest degree of hurt by deliberately leaving others out of activities.

Psychological profiles of bullies universally indicate that they suffer from low self esteem, despite their bravado. Victims, also fall prey to low self esteem and in many cases resort to bullying themselves in order to break the cycle. Schools have anti-bullying policies in place and discipline strategies to prevent such behaviour, but a bully becomes a master of disguise and their behaviour often continues despite the best efforts of teachers and parents to minimise its chances.

Letting a child who is being bullied, know that they can do something about it, is the first step. Giving them strategies for action, rather than reaction is empowering. Most kids relate to the experiences of others and books on both the topics of friendship and bullying are a great way to share ideas and deliver positive messages. Two Australian authors who have written about friendship and bullying are Sue Whiting and Susanne Gervay .

Sue’s book, Elephant Dance touches on the topic of ‘falling out’ with friends and Susanne’s, I am Jack delivers a first hand account of her son’s experiences with bullies. Both make for great reading.

Elephant Dance. Koala Books.

Written by Sue Whiting and illustrated by Nina Rycroft, Elephant Dance is a touching story about the universal theme of friendship. Two elephants, Hugo and Millie are the central characters. They share a special friendship until a difference of opinion causes a commotion and a parting of the ways. Pride gives way to loneliness and when Hugo’s solo journey lands him in danger, Millie comes to the rescue. The result is a happy reunion, with both characters wiser for the experience.

The text has great rhythmical appeal, which lends itself beautifully for reading out aloud with young audiences joining in.

Bright colours on the cover are continued throughout the story delivering a visually attractive and appealing journey. The target audience is 3 – 8 year olds, but the message of friendship being a cornerstone of our lives, makes it a suitable read for a variety of age groups.

I Am Jack. Angus and Robertson

Susanne Gervay’s novel, I Am Jack, is based on the experiences of her own son at the hands of schoolyard bullies. It addresses the isolation and desperation of the victim of bullying, yet contains just the right mix of humour to keep it entertaining and enthralling for its intended audience – children.

The book highlights to both parents and educators, how changes in a child’s behaviour are the initial warning signs. It also serves as a timely reminder of how busy lives can sometimes leave us unaware of important signals being sent from those closest to us.

The story is woven around Jack’s family life with his single mum, Rod, his soon to be step dad, his sister Samantha, his nana and his best friend, Anna.

I Am Jack, addresses the problem of bullying in a warm, uplifting way and more importantly extends a message of hope to any child who falls prey to this insidious practice.



SOURCE:
http://www.naturalparenting.com.au/flex/articles/7961/friendships-an-important-part-of-childhood.cfm