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Problem Solving with Bar Models

November 1, 2018

Christopher Coyne
National Education Consultant

I recently came across a quote by the English writer and philosopher G.K. Chesterton: “It isn’t that they can’t see the solution. It’s that they can’t see the problem.”

Problem-solving in math class requires more than procedural skills. Students must first understand the problem before attempting a solution and explaining their thinking. This can present a challenge in our classrooms. Bar modeling can be a tool to help teachers and students meet this problem-solving challenge, and is often considered the hallmark of Singapore Math®. It can help students “see” the problem by representing known and unknown quantities, and their relationships – sounds sneakily algebraic, doesn’t it?

Let’s consider the following problem:

On Halloween, Logan ended up with 7 fewer pieces of candy than Kristin. They have 55 pieces of candy altogether. Find the number of pieces Logan has.

Students who don’t understand the problem may mistakenly subtract based upon the word “fewer”.

For those of us that remember algebra, we can represent the problem this way:        

2x + 7 = 55. Algebraically, we would isolate the variable and solve for x (x = 24. Logan has 24 pieces of candy). Although perfectly acceptable, the algebraic solution is abstract and developmentally inaccessible to most elementary students, yet they still need to solve this type of problem. Bar modeling can make it more accessible.

Let’s draw a bar for Logan and a bar for Kristin. At this point, I’m not quite sure how long to draw the bars but I know that Logan’s will be shorter because he has fewer pieces of candy.

Now let’s use the information in the problem to label the model. I know that Logan has 7 fewer and they have 55 altogether.


 The problem asks us to find the number of pieces Logan has so we label our question mark and the model is done.

The bar model has helped us organize the information from the problem but we still don’t have an answer. Students still need the skills from previous lessons and by now have many strategies to solve 55 – 7 and 48 ÷ 2.

Bar models also help students gain a deeper understanding of the operations they may need to use rather than simply relying on keywords, usually taken out of context. Singapore Math® sets the stage in kindergarten and first grade for bar modeling by developing number sense, fluency and part-whole thinking. From second grade onward, the bar modeling strategy helps students organize information – it becomes a graphic organizer for math – in order to “see” the problem to better understand it and become better problem solvers.


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Back to School with Singapore Math® – Let Today Be the Start of Something New!

September 18, 2018

There are many places of employment that promise a new start with New Year’s Day or the start of a new fiscal year but more often than not that new start is fleeting and the same old ruts soon reemerge.

Education is unique in that we truly start anew with each school year. It is full of opportunity to get to know another group of students, reacquaint ourselves with colleagues and administrators – or perhaps get to know new ones. It is also a chance to revisit our classroom management procedures and implement new best practices in our classrooms. It is our time to shine with each new teachable moment. As teachers, we know that it is the procedures and practices along with appropriate responsibility and consequences that aid in our sanity throughout the new year, and we need to begin with them from day one.

When it comes to Singapore Math®, the research is clear – the use of manipulatives is non-negotiable. With a new school year, we need to establish the structure for their use with our new group of students. Manipulatives need to be readily available in your classroom. If we ask students to “hold that thought” so we can go look for manipulatives, we have lost a teachable moment. As educators, we need to consider how we will create the opportunity for students of all levels – those who may be struggling with a particular topic, those who are moving towards mastery with a particular topic and those who could benefit from enrichment with a particular topic – to make use of manipulatives to see and explain math.

Another teaching approach which should be established at the start of the school year is the use of productive struggle to help students build confidence and learning persistence.  Singapore Math® offers ample opportunity for students to discuss, justify and analyze their answers and strategies – as well as those of their peers – through the use of problem-solving.  The Singapore Math® lesson gives students an opportunity to discover the process needed to solve a given problem through discourse, exploration, and revision. As teachers, we must allow wait time as we encourage student participation, ask students to provide additional methods and probe student thinking as they productively struggle through problems.  Sometimes this wait time can be difficult but as educators, we must be willing to let students think and work through math problems before swooping in to help guide them.

Setting the procedures and the expectations from day one sets us up for a positive year and sets our students up for success in math.

Christopher Coyne
National Education Consultant

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How and Why to Facilitate Differentiation in Your Math Classroom

September 6, 2018

Christopher Coyne
National Education Consultant

There is much written about differentiation in the math classroom. But at the core of it all, there seems to be a central idea: a student-centered classroom managed by a teacher who knows the varying learning needs of his or her students and addresses those needs appropriately.

Easy, right? Well, as Mark Twain said, “If talking were teaching, we would all be smarter than we could stand.”

We kept this in mind when developing Singapore Math®, which uses a problem-solving approach and incorporates opportunities for multiple representations—both of which provide the context for differentiation. Let’s consider this adding with regrouping problem:

Can you find one way? Can you find more than one way? How many ways are there, and why? Think about this problem with your students. Which students would you ask for one way? More than one way? How many ways?

With Singapore Math®, differentiation often becomes about what questions to ask students rather than developing different tasks. Perhaps your struggling students find one way, while your on-level students find more than one way, and the higher-achievers for this lesson find how many ways and why. Students must learn how to break concepts down and, just as important, how to build them up. Differentiation is as important for advanced learners as it is for struggling learners.

An opportunity for multiple representations helps students conceptualize the math while also helping to develop number sense. Let’s consider subtraction. If students are simply taught to put the “larger number on top and the lower number on the bottom” to subtract, it will likely lead to subsequent issues. Students may end up incorrectly subtracting decimals (not to mention integers) like this—after all, the larger number is on top:

Through the use of multiple representations—including concrete manipulatives, pictorial representations, and abstract symbols—students can begin to visualize the math (in this subtraction problem, place value) and develop strategies which lead to them developing number sense, persistence, and confidence in math class.

Interested in learning more about Singapore Math®, which aims to facilitate differentiation in the math classroom? Register here for our webinar, “Differentiating and Small Group Instruction in Singapore Math®,” presented by Terry Goldfischer and Christopher Coyne on Sept. 19 at 4 to 5 p.m. ET.

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Smart Students + Smart Schools = Smart Nation

March 15, 2016

Should U.S. students live in a Smart Nation?  Currently, Singapore is on their way to becoming the first nation in the world to be a “Smart Nation”.


Singapore likes to be first and rank first.  For more than 20 years now, Singapore has consistently ranked first (or near first) in mathematics and science.  While waiting at the airport  last week, my CNN app sent me an alert that Singapore is now first on the list of the World’s 10 Most Expensive Cities To Live In1.  Being first or being in first place seems to be part of their DNA.

Singapore’s government, through their office of The Infocomm Development Authority (IDA), wants to make the country the world’s first true Smart Nation2.  Their slogan is “E3A”: Everyone, Everything, Everywhere, All the time.  What makes a nation “Smart”? – Technology.

Like most developed countries, Singapore has big problems. Transportation problems, population problems, security problems, healthcare problems and many others and they are convinced that technology is the pathway that will enable them to develop and grow infrastructures and technology capabilities to help its citizens, businesses, and government solve their nation’s problems.

How is math class related to all this?  Well, a Smart Nation needs Smart Students.  We need to make a distinction, “Smart” is not the same as “smart”.  Carol Dweck, author of Mindset, might argue that students can always grow “smarter”3.  In this context, a “Smart” student is a student who is able to use technology to innovate and solve problems.  A Smart Student uses logic and thinking skills through technology.  A Smart student surely gets smarter.

Those familiar with Singapore Math textbooks in the U.S. are also familiar with the Singapore Math’s Framework Pentagon that specifically has problem solving as the focus of mathematics instruction.  These students, with their teachers’ help are becoming smarter students and better problem solvers.  Incorporating technology to the math class can further develop these skills.

Singapore Math Framework

Recently in the news in Singapore, I read an article about young students signing up for coding workshops and classes offered by various companies4.  The goal is not to develop programmers but to train them in logic and clear thinking.  Becoming a programmer is not such a bad idea either and according to the App Economy (apparently a real term) more than 627,000 jobs5 have already been created and growing rapidly here in the U.S.  Parents in Singapore are sending their children in droves to these types of “enrichment” coding classes because they are recognizing two things: 1) technology is the present and future for their children and 2) developing logic and thinking skills directly benefits their current schoolwork.

Another important component in achieving a Smart Nation is the need for Smart Schools.  Learning in the 21st century demands technology.  School districts everywhere are planning and implementing these plans on improving and developing their technology infrastructures.  However, having high-tech alone is not enough.  Schools everywhere are reimagining teaching by using technology.  One such impressive initiative can be found in New York State’s “NY Smart Schools Commission Report”6  The Keys to Success for Achieving a Smart School is very helpful and critical is key number 5: Provide high-quality, continuous professional development to teachers, principals, and staff to ensure successful integration of technology into the teaching and learning experience.  Is your school a Smart School?

U.S. schools across the country have made significant infrastructure upgrades.  It was rare only a few years ago to see schools that had more than just a few computer workstations in the classrooms.  Soon after that, carts of iPads or Chromebooks became normal.  Now students bring their own mobile devices.  However, in many places there is a sense that students are and have been ready for Smart Schools for a while now and we are the ones trying to catch up.  For example, I recently visited a school in Texas that was using Math Buddies (a Marshall Cavendish Education digital program) and I was blown away by how fluid they were in their ability to multitask between talking and helping each other, using paper and pencil to do scratch work, dragging and dropping on screen, typing and giving each other high-fives because they solved a problem correctly. A Kindergarten teacher confessed that one surprising challenge had nothing to do with the program but with their hardware as her students had to be taught how to use a computer mouse because instinctively, her students wanted to just touch and swipe the screen to get the program to do what they wanted it to do.  In some cases, it is us, the adults who have to do the catching up.  Young students have been ready to be part of Smart Schools for a long while.

Personally, my own 13 year old son has recently shown interest in being part of a Hackathon.  Suspecting it might be some nefarious “-athon” I looked it up and realized that a hackathon is a marathon computer-programming competition.  Again, as an adult, I am the one needing to catch up.

The more Smart Students we have in our Smart Schools the closer we too are to becoming a Smart Nation.

Do you teach Smart Students?  Is your school a Smart School?  Do you teach in a Smart Math Class?  We’d love to hear about your experiences in the classroom with technology.  Be sure to leave a comment back at the LinkedIn blog area.  Share with us the amazing logic and thinking skills your students have demonstrated using tech.  Tell us a story about the Smart Students you work with.

Share your thoughts on this article or any other Singapore Math topic by joining our LinkedIn community.

By Hoover Herrera
Singapore Math® expert


  1. CNN app. (accessed on March 11, 2016)

  1. “IDA wants to make Singapore a Smart Nation. Here’s what you need to know”. Article author: Michael Tegos. (accessed on March 3, 2016)

  1. ”Even Geniuses Work Hard”. Author Carol S. Dweck (accessed on March 4, 2016)

  1. The Straits Times, Singapore. Author Lester Hio. (accessed on March 10, 2016)

  1. The Atlantic online newsletter. “The Secret Lives of Hackathon Junkies”. Author Meredith Broussard. (accessed on March 1, 2016)
  2. Smart Schools New York Commission. (accessed on March 11, 2016)



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