## Guest Post – Problem Solving: The Heart of Singapore Math

This article originally was featured in the fall 2018 issue of St. Anne’s-Belfield School’s Perspectives magazine and is republished with both the school and author’s permission.

The author, Sara Kronstain, has almost a decade of experience guiding elementary and middle school mathematicians to become critical thinkers and problem solvers.  She teaches fifth-grade math and is the Kindergarten-6th Grade Math Department Chair at St. Anne’s-Belfield School in Charlottesville, Virginia.

## Problem-Solving: The Heart of Singapore Math

When I was in school, my math classes were typical of what one would expect a “traditional” math class to look like. I remember sitting in my elementary and middle school classes, watching as my teachers modeled problem after problem. The class would listen and then practice many of the same types of problems in our notebooks. While this type of teaching may achieve the immediate goal of learning a mathematical procedure, it does not guide students to reach an integral part of learning mathematics: problem-solving (Cai & Lester, 2010).

Singapore Math is comprised of a framework with problem-solving being the center of learning mathematics. This framework is built around five key components – metacognition, process, attitudes, skills, and concepts – all being of equal importance in developing mathematical problem solving in students. Whereas traditional math classes may place primary importance on developing skills and concepts in students, the additional three components of metacognition (self-regulation of learning), process (reasoning, making connections, and applying knowledge), and attitudes (perseverance, confidence, interest) are all key to developing critical thinking and problem solving skills in students (Ministry of Education Singapore, 2006).

A typical Singapore Math lesson is taught with a concrete-pictorial-abstract approach. Where many of my lessons as a math student began in the abstract stage (solving equations), the concrete and pictorial stages allow students to create and solidify their own understanding of a topic. The concrete stage refers to using hands-on materials to model a mathematical situation.  The pictorial stage consists of diagrams and other visuals, thus building students’ learning in a tangible way (Maths No Problem!, 2018). The concrete and pictorial stages allow students to understand why math works the way it does before learning the procedure of how to solve using an algorithm.

Most Singapore Math lessons begin with an anchor task, allowing students to explore these three stages. The anchor task is a question that allows students the chance to deeply explore a topic and develop multiple methods for solving a problem (Ban Har, 2013). Let us say, for example, a group of fifth graders were posed the problem, “The distance of a race is 3km. Lily ran two-fifths of the distance. How many kilometers did Lily run?”  Students would be given the opportunity to freely explore this question by using manipulatives such as fraction bars, fractions circles, or paper (for folding) along with writing materials. Here are a few examples of possible student responses to this question:

Add ⅖ + ⅖ + ⅖Students may use fraction bars, fraction circles, or pictures.  Students become familiar with the phrase “3 groups of ⅖”.

#### Method: Bars

Three boxes are each split into fifths. Two of each of the fifths in all three boxes are shaded in.  The shaded parts are added together.

#### Method: Bar Model

A bar with the length of 3 wholes can be split into five parts.  Each part has a value of ⅗. Then add ⅗ + ⅗ .

Three boxes are split into 5 equal groups, first by placing one half in each group. Then, split the leftover half into five parts (tenths). Each group will have one half of a whole and one tenth of a whole. Combine two groups by adding two halves to two tenths.

In this example, the repeated addition method reinforces addition with fractions, while the last method has students thinking about and manipulating fractions in a much more complex way.  Thinking back to the five key components of Singapore Math, students in this example are refining their process of learning operations with fractions by making connects across operations.  It is powerful that these responses are coming from students, as they are building their understanding of math through collaboration with their peers. In sharing methods, listening to other’s methods, and processing others methods, students are also developing their metacognition. This question could also be modified and challenge students to problem solve in an even deeper way. “What if the total distance was ½ km?  What if the total distance was 3 ½ km?”  Students can then go back to the concrete, pictorial, and abstract stages and continue to build on their problem-solving abilities.

At the end of the day the primary purpose of this math lesson, or any math lesson for that matter, is not simply to learn how to multiply fractions by a whole number. The most important takeaways are the critical thinking, questioning, collaboration, and problem-solving that happens among students. Teachers are not preparing students to go out into a world where they will simply be asked to recite an algorithm. While a goal is for each child to develop a deep love of math, the biggest hope is that students learn to ask questions, logically think through problems, and make sense of the world around them.

#### References

Ban Har, Yeap.  (2013, June 13).  Singapore Math at the Blake School, Hopkins, MN.  Retrieved from http://banhar.blogspot.com/search?q=anchor+task

Cai, Jinfa, & Lester, Frank. (2010, April 8).  Problem Solving. National Council of Teachers of Mathematics.  Retrieved from https://www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Problem-Solving/#brief

Kaur, Berinderjeet.  (2018, March 29).  Building the Maths house: Singapore’s curriculum framework.  Oxford Education Blog.  Retrieved from https://educationblog.oup.com/secondary/maths/building-the-maths-house-singapores-curriculum-framework

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## Announcing: Jumpstart Your Singapore Math® Instruction Workshops for 2018!

Back by popular demand!

We are pleased to announce the return of Jumpstart, an intensive, two-day workshop for current and potential users of Primary Math and Math in Focus, as well as any teacher interested in incorporating these techniques into their own classroom, regardless of current curriculum. If you are:

• new to the Singapore approach to math instruction…
• needing a refresher to boost your math teaching skills…
• wanting to incorporate the best practices from Singapore into your current curriculum…or
• curious about the reasons for Singapore’s remarkable success…

…then this workshop is for you!

Click here to get all of the details on this exciting program!

#### Location and dates currently available:

Tulsa, OK | July 23 – 24, 2018:

Fort Collins, CO | July 26 – 27, 2018:

Minneapolis-St. Paul, MN | July 30-31, 2018

Do you want to be notified when a Jumpstart Your Singapore Math Instruction is scheduled near you? Fill out the form below:

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## Structuring the Math Day

One of the questions I get most often is:

##### How do I use the materials with my Singapore Math curriculum and fit it all into an hour math block?

First off, kudos to your school for setting aside an hour math block for your youngest learners! Through math instruction, students will gain the skills and thought processes necessary to solve problems. Math needs to be given a priority in the schedule. Following is one of my favorite quotes from Dr. Yeap Ban Har, author, and contributor to several Singapore Math style curriculum.

“We are not teaching math. We are teaching thinking through the medium of math.”

##### What should I include in my lessons?
• Ongoing cumulative review
• Direct instruction
• Guided practice
• Independent practice
##### How much time should I spend on each component?

10 minutes – Ongoing Cumulative Review
20 minutes – Direct Instruction
30 minutes – Guided and Independent Practice

##### What does each component consist of?
###### Ongoing Cumulative Review (10 minutes)

According to Steven Leinwand, in his book Accessible Mathematics: 10 Instructional Shifts that Raise Student Achievement, in every classroom there should be signs of:

A deliberate and carefully planned reliance on ongoing, cumulative review of key skills and concepts.

As you teach concepts, you will want to include them in your ongoing cumulative review. With such an emphasis on mental math strategies and the development of number sense, mental math should play a major role in your daily review.

Mental Math can be practiced through the use of:

Direct Instruction (20 minutes)

• Teacher directed (follow the plan in the Teacher’s Guide)
• Through student exploration (also known as, an Anchor Task)

Guided Practice (30 minutes combined with Independent Practice)

• Textbook problems can be worked:
• Whole group answering problems on individual whiteboards,
• With partners working through problems together, or
• Individually

Independent Practice (30 minutes combined with Guided Practice)

• Workbook problems
• As home enjoyment
• As classwork
• Fluency practice

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## Throwback Thursday – It can’t all be Singapore Math…

Over the summer, we thought it would be fun to run some of the most popular posts from the past. Here’s a look at some misconceptions around Singapore Math and Common Core Standards. When I re-read a post from the past I always take away something different because I am in a different place with my own experience. Perhaps you are as well!

### It can’t all be Singapore Math…

###### Originally published 12/29/2014

This tweet posted by the National Council on Teacher Quality (@NCTQ) caught my eye:

Now, I’ve heard decomposing called “branching” but can’t remember ever seeing this in a Singapore textbook. Where did this problem come from?

It’s nice that NCTQ recognizes Singapore’s Math as “tops in the world.” But it’s discouraging to see methods and terminology that are not a part of the Singapore curriculum attributed to it. Especially in the context of the nasty debate about CCSS. And especially since Singapore’s math curriculum–with its rigor, coherence, and focus–is often cited as a basis for more rigorous standards, including CCSS.

The problem posted is based on the concept of “Number Bonds,” which calls for students to decompose numbers (this is the term used in Singapore and in all major Singapore Math® textbooks distributed in the U.S.). Below, I’ve posted some examples of how this concept is presented in Singapore Math® series available in both the U.S. and Singapore.

This matter points to my BIG concern: As publishers and others adapt Singapore’s Math for the American market, new approaches creep in. These often are not based on the curriculum that helped Singapore’s students go from mediocre to best in the world in a dozen years. I’ve written about this in my comparison of Singapore math textbook series available in the United States.

So my plea to NCTQ: please use examples from an actual Singapore mathematics text when citing the components that make it so successful. And feel free to ask if I can help you find those examples.

#### Number Bonds problems in Singapore Math® textbooks

Here are some materials covering Number Bonds and “decomposing” numbers from actual Singapore textbooks:

From My Pals are Here, the most-used materials in Singapore:

From the U.S. Edition of Primary Mathematics, available in North America since 2003:

From the Common Core Edition of Primary Mathematics, released in the U.S. market in 2014:

And finally, from Math in Focus:

UPDATE:

Ugh! One more similar tweet from NCTQ.

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## Event Recap: Jumpstart Your Singapore Math Instruction

[Several teachers from Minnehaha Academy in Minneapolis attended Jumpstart. We are stunned and saddened by the news of the explosion at Minnehaha’s Upper School and our thoughts are with the Minnehaha community and families of the victims.]

Beth Curran and I had wonderful time hosting Jumpstart Your Singapore Math Instruction, our recent two-day workshop (and first self-produced event) in Saint Paul, Minnesota.

Our goal: to offer an intensive learning opportunity covering all the essential elements of Singapore’s highly-acclaimed math program, including strategies, number sense, model drawing, and curriculum.

We were thrilled to be surrounded by such an enthusiastic group of co-learners, including teachers from many of the region’s finest schools and some from as far away as California.

What did teachers say about their experience?

I couldn’t imagine teaching this year without this class. It was amazing!

-Joanne Pilon, 6-8 Math, Holy Family Academy

Wonderful experience! Amazing! Thank You!!!

-Sarah Gerlach, 2nd Grade Teacher, St. Raphael Catholic School

OUTSTANDING!! After all of our training, it’s all making sense now. I LOVE IT!!

-Kim Schafer, 4th Grade, Breck School

Amazing and informative CEU that I can apply instantly to my teaching practices.

-Sharleen Blanco, Kindergarten Teacher, Excell Academy

Very informative, fun, and safe space to learn about and practice Singapore Math. Thank you!