## Assessing Bar Model Solutions

It’s important in a Singapore style program for students to understand bar model drawing as a tool to help visualize relationships between the known and unknown in a word problem. It helps students see the algebraic structure in problems in a more concrete manner.  Developed by a Primary Mathematics Project team of the Curriculum Development Institute of Singapore in the 1980’s, the Model Method for problem solving (as it’s known in Singapore) was designed as a pictorial stage to help students learn abstract mathematics.

The model method has been incorporated into the Progressions documents for the Common Core State Standards for Math (CCSSM) as a “tape diagram”. The CCSSM glossary defines a tape diagram as:

“A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length model.”

As such, bar models are becoming ubiquitous in elementary schools. Books have been written about Singapore’s model method, my favorite being  The Singapore Model Method for Learning Mathematics:

There’s plenty to be covered with bar models, and I thought I’d share answers to the most commonly asked questions I get from teachers on model drawing:

How do you get kids to draw models when they already know the math?
How do you assess them?

When models are introduced in Singapore textbooks, the problems are very simple and students will typically know what operation to use. The first part whole models in Primary Mathematics, Common Core Edition are in the 3A textbook:

And the first part-whole model lessons in Math in Focus are in 2A:

Those are pretty easy word problems and most kids are waving their hands, with that “I KNOW the answer” look on their faces. It’s important to reiterate to students, “I’m sure you know the answer and even the equation to get there. What we’re learning today is a new way of drawing a model for our word problem so that we can work with more challenging problems in third grade. Problems like”:

A father shares \$60 among his three sons and one daughter. If the daughter gets twice the amount that each son gets, how much money does his daughter receive?

or from 4A

We’ll come back to that one.

When assessing word problems, the solution method is a significant part of the answer. Students should expect to show their work to get full credit for a problem. As a teacher, it’s important to keep in mind that there is no one-way to draw a bar model.

A word problem might consist of points for the method of solving as well as points for correct computation and answers. To encourage students to draw and become proficient with bar models, I have used a rubric for assessment:

• 1 point for a representative diagram – Does the model make mathematical sense?
• 1 point for correct labeling, including the “?”  to represent what is unknown.
• 1 point for computation on the first step and, if more than one step…
• 1 point for computation on the second or more steps
• 1 point for a correct answer in a complete sentence

Let’s look at an example. This is a student-written two-step word problem from third grade:

Does the bar model make sense? +1 point

Are the bars labeled? Are question marks in the correct places? +1 point

Is the computation to figure out the value of one unit correct? 384 ÷ 6 = 64 +1 point

How about the computation on the second step? 64 x 5 = 320 +1 point

Is the answer in a complete sentence? +1 point

So, what’s the deal with the 64 ÷ 6 equation? A byproduct of requiring students to show their work is that oftentimes, they will leave work on the paper, just in case. As a teacher, I love that this provides me with some insight into this student’s thought process:

I’m pretty sure I know what I’m doing, but I still am a little confused.

Which is exactly what the student said when I asked her about the extra equation. She realized her answer couldn’t be right because:

I ended up with 14 remainder 4 meatballs, and how could that happen?

Without a system to evaluate, you end up with:

I’m drawing boxes because I’m supposed to… I don’t get it.

This model doesn’t make sense. The student’s computation is slightly off, but she knows she needs to add. A teacher can also see from this model that more re-teaching and practice is necessary for this student and probably others in the class.

As students progress, the rubric may change. At the end of a unit/school year, the same problem might be worth three points; one apiece for diagram, computation, solution. I often tell students, “You don’t need to draw the model if you feel you understand the problem, however, you will be graded either all correct or all incorrect. Without the models I can’t give you partial credit for your thinking.”

Most students draw them anyway as a way to check their work. Some just start to see them in their heads. Here are a couple of examples from the end of a fractions unit in 5A

How would drawing a model have helped this student?

## Oxford University Events featuring Singapore Maths (and Me!)

Photo: Pablo Fernández/Flickr Creative Commons

I’m thrilled to announce that I’ll be speaking at not just one, but two different international conferences at Oxford University in England this June. And I am deeply honored by the invitation to present a Keynote address at one. Here are the details:

### researchED Maths and Science Conference

Saturday, June 11th, 2016 — Oxford’s Mathematical Institute

reseachED are holding an inaugural conference in Oxford aimed at primary and secondary maths and science teachers. I’ll be on a panel with Sue Lowndes and Dr. Fong Ho Kheong that will explore the implementation of the Singapore approach in non-Singaporean countries; what do we know, what we have learned and what should we do going forwards. I’ll be the representing U.S. contingent.

### International Forum: Implications for UK practice in using the Singapore approach to teaching and learning in mathematics

Monday and Tuesday, June 13 – 14, 2016 — St. Anne’s College, Oxford

I will be speaking at two sessions at this conference. On Day 2, I’ll present a keynote entitled: You’ve adopted the Singapore approach to teaching mathematics – now what? Singapore maths curricula have a proven track record for meeting standards in mathematics. They can, however, be seen as a departure from prior curricula that teachers have used. This session will explore what teachers and senior leaders should consider throughout the implementation process.

I’ll also head up a workshop session, Filling in Knowledge Gaps: Critical lessons across the year groups.  Upper KS2 students (that’s ages 8 to 11 to Americans) frequently lack the foundations required in order to successfully follow the Singapore curriculum. In this ‘straight from the classroom’ session, we’ll discuss the critical lessons and concepts students must master before jumping into their year-level content.

I understand that registration is brisk at these sessions. Hope to see you there!

## New Resources, New Workshops

Over the last ten years, I’ve collected a lot of materials that I love to use in the classroom. Teachers are always asking for new ideas and games so I’ve created a new page: Favorite Printable Math Resources. Feel free to email your favorites to be included on this page!

2016 dates for my BER seminar, “How to Use the Best Strategies from Singapore Mathematics to Strengthen your Math Instruction” have just been released. Updated with March dates on 10/14/2015!

This overview of Singapore Math® strategies will put your students on the road to success with number sense, computation and problem solving. (Plus you a get a comprehensive resource handbook to take back to your classroom!)

Want to see Cassy, but a city near you is not on this list? If so, complete the form on the Bring Cassy to my School/Area page or send her an email.

## Fall 2015 dates for Model Drawing BER Seminar

Best. Seminar. Comment. Ever.

Here are the just-released Fall 2015 dates for my BER seminar “Boost Students’ Math Problem-Solving Skills Using Bar Models, Tape Diagrams and Strip Models (Grades 2-6)

Save the date!

A whole day of problem-solving with Bar Modeling, Tape Diagrams and Strip Models PLUS that handbook for your own home enjoyment! (- with the answers and fully worked solutions!)

——————————————————————————————————————————–

“How to Use the Best Strategies From Singapore Mathematics to Strengthen Your Math Instruction” (BER) will have 10 dates in spring of 2016 – I’ll update when available.

Your city not on the list? Contact me and I can bring my Singapore Math® workshop(s) to your school or district – email Cassy (at) singaporemathsource.com

## Our Journey to Singapore: A Singapore Math Adoption Success Story

Beth Curran

For some time, I’ve wanted to share stories of schools that have successfully implemented a Singapore Math curriculum.

To present the first such case study, I asked my colleague Beth Curran to summarize the adoption process at St. Anne’s-Belfield School, an independent Pre-K to 12 school in Charlottesville, Virginia.

#### Our Journey to Singapore

by Beth Curran
Preschool – 6th Grade Math Department Chair, St. Anne’s-Belfield School
Singapore Math Teacher and Trainer

It all began with a strategic plan.  In 2011, St. Anne’s-Belfield School released its 2011-2016 Strategic Plan.  The first of six goals focused on teaching and learning in the 21st century.  Key elements to this goal included teaching with depth rather than breadth, teaching critical thinking and problem-solving skills, improving the quality of our computation, and ensuring that our pedagogy reflects researched based best practices.  The Action Plan that followed gave direct mention to Singapore Math as a curriculum to explore.

While teachers felt strongly that the students were leaving our Lower School (grades Kindergarten through four) very well prepared for Middle School (grades five through eight), we had to ask a tough question; could we be doing better?

#### Why Singapore Math?

As the Lower School Math Coordinator at the time, I was charged with taking a critical look at the Lower School’s current math curriculum and learning all I could about Singapore Math.  The more I learned, the more I was convinced that Singapore Math would be a great match for us.  It was almost as if the Strategic Plan was written with Singapore Math in mind.  The curriculum teaches concepts to mastery, focusing on depth rather than breadth.  Critical thinking and problem-solving are embedded within the curriculum, not taught as a stand-alone unit.  Concepts are introduced, practiced, and applied immediately to solve problems.  Computation and numeracy are also a major focus.  Check, check, and check!

Learning Village at St. Anne’s-Belfield School

Not all of the homeroom teachers were as enthusiastic as I was.  It was a daunting task convincing them that learning a new math curriculum, on the tails of learning a new writing curriculum, was a good thing.  St. Anne’s-Belfield’s Head of School, being the visionary that he is, saw an opportunity to not only implement a new math curriculum, but to change the way math instruction is delivered at the Lower School level.  If we were going to ask our teachers to become Singapore Math specialists, why not hire and train dedicated math teachers?  And that’s just what he did.  Four math teachers were hired to deliver math instruction and these dedicated math specialists would co-teach math with the homeroom teacher taking on a supporting role.  This had an added benefit of cutting our student to teacher ratio in half during math class.

With the faculty in place and the Primary Mathematics materials ordered, we set out to train our dedicated math teachers in Kindergarten through sixth grade.  We contracted with Cassy Turner, Singapore Math Specialist and Trainer to work with our math teachers for an intensive one-week boot camp.  We learned the ins and outs of mental math and the bar model.  We asked questions, practiced, collaborated, practiced, designed an implementation schedule, and practiced.  Cassy’s enthusiasm and extensive knowledge left us feeling confident to tackle the upcoming year.  We knew professional development was crucial to a successful implementation and with that in mind we continued our relationship with Cassy throughout the year.  She made three more trips to the school, observing and teaching lessons and providing her guidance to keep us on track.

#### Successes and Challenges

Fast forward to today.  We are now a year and a half into our implementation. Our students are stronger problem-solvers than ever before.  Their computational skills have shown marked improvement.  Their overall sense of number and place value has increased.  Our students are confident and persevere through challenging problems.

We have done a lot of things really well.  We understood and placed value on professional development.  This is not a curriculum that can be picked up and taught from the Teacher’s Guides.  Most teachers did not learn math the way that a Singapore Math curriculum is taught.  Training is key.  If not trained, teachers will revert to teaching math the way they learned it.  Having a successful plan for ongoing professional development is critical to a successful implementation.

We put value on mathematics instruction at the Lower School level.  We saw the need for math specialists and took a huge financial risk to improve our instruction.

We implemented the curriculum in Kindergarten through sixth grade.  We felt so strongly about the benefits of the curriculum that we knew that even one or two years of exposure would be better than none.  This has been one of the most challenging hurdles of our implementation.  We worked with Cassy to anticipate and develop a plan for “back-teaching” missing skills.   In grades three through six, this plan guided us through our first year and fortunately, Kindergarteners through second grade students benefited from needing very minimal “back-teaching.”  Developing a relationship with a knowledgeable Singapore Math consultant is crucial.

If there was an area for improvement, it was parent communication and education.  We hosted a parent night early into the school year to give parents an overview of the curriculum and a brief introduction to some of the components that are unique to Singapore Math.  That wasn’t enough.  Parents didn’t learn math the way their children were now learning it. The focus of Singapore Math is to develop conceptual understanding before learning the mathematical steps or procedures.  Parents need to understand and support the school in teaching math this way.  Parent education is not an option; it is a requirement of a successful implementation.  In our second year, we designed a plan for parent chats spread throughout the year with topics including fact practice, mental math strategies, and bar modeling as a tool for problem-solving.  Your professional development provider or consultant can assist you in designing a parent education program that meets the needs of your school.

Our journey continues and our students are stronger math students as a result.  The first year was clearly the most challenging.  Our commitment to professional development, perseverance, and acceptance of this unfamiliar approach to teaching math has guided us and we are confident that each passing year will continue to confirm the benefits of teaching a Singapore Math curriculum.