## 4,000 Teachers, 100,000 Students: Celebrating 7 Years with BER

In 2008, I left teaching in the classroom to champion Singapore Mathematics and expand its reach to elementary schools and children everywhere. The following year, the Bureau of Education and Research (BER) gave me an amazing opportunity to pursue this goal by presenting Singapore Math workshops throughout North America.

Some attendees at my BER seminars came with prior knowledge about the Singapore curriculum, but a bigger number were being introduced to Math from Singapore for the first time.

At a Seattle workshop earlier this year, BER’s Mark Ita surprised me (and other attendees) by presenting me with a handsome plaque, which read, in part:

In Recognition of Your Distinguished Teaching and Your Outstanding Contribution to the Education Profession

#### 4,000 Teachers, 100,000 Students

The stats scribbled on a Post-It Note on the back of the plaque included some tangible data to support this statement:

• 165 Seminars
• 4,000 Teachers
• Over 100,000 Students

Cassy with BER’s Mark Ita

It is highly satisfying to know that I have impacted this number of teachers and students through my BER presentations. On the other hand, the National Center for Education Statistics reports that there are about 35.2 million Pre-K to Grade 8 students in the United States. Clearly, there is much more work to be done!

I am very grateful to BER for giving me the opportunity to present Singapore Math workshops on their behalf over the past seven years. Sincere thanks to Rich, Boyce, Mark, Nargis, Lisa and the entire travel logistics team, and the dozens of project managers who have provided encouragement and support along the way. Thank you so much!

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## Bar Model Solutions – by Students

After the post on Assessing Bar Model Solutions went up, Beth Curran sent a message: “We just did that problem!” She agreed to share some student work:

And when the students didn’t draw a model:

I see this as a comparison problem:

5 units -> 125 students
1 unit -> 125 ÷ 5 = 25
7 units for boys -> 7 x 25 = 175 boys in all

(That’s the Thinking Blocks Model Drawing tool that allows you to insert your own word problems and solve – or you can use the pre-made questions!)

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## 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?

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## 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!)

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“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

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## You Can Bar Model Anything!

A fourth grader at a school I worked with this year included this on a Christmas card for her teacher.

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