As a teacher, this is a glorious time of year, but also one of worry. I worry about my students. I worry about those who needed extra support throughout the year understanding and retaining math concepts. How will they fare next school year? Will they regress over the summer months if they don’t do any math work?

There are three categories of students who benefit most from summer math work:

- Those who have struggled all year and maybe never quite achieved mastery on those critical grade level concepts,
- those who easily forget concepts, and
- those whose math confidence could use a boost.

With a Singapore Math program, there aren’t many ready-made options to pick up at the local bookstore. Books that are available focus heavily on procedural understanding rather than underlying math concepts. So what’s a teacher to do?

Aside from recommending tutoring, I have found a couple of options that seem to meet my needs as a teacher and the needs of my students.

For those looking for a paper and pencil option, I recommend the *Extra Practice* books from Singapore Math’s Primary Mathematics series. Students should work at the grade level just completed (a rising 3rd-grade student should do summer work in the 2^{nd} grade *Extra Practice *book).

The *Extra Practice* books offer parents and/or tutors “Friendly Notes” at the beginning of each unit that explain how to re-teach concepts in a way that is familiar to the student. The notes are followed by practice pages that give parents sample problems appropriate for practicing the concepts and the student an option of working through problems independently. Best of all, they include an answer key in the back so parents can check work and students can re-work problems, if necessary.

These books are written to cover a year’s worth of concepts; I am by no means suggesting that a child complete the entire book over the summer. Teachers recommending this book will need to tailor the tasks to meet each student’s needs. This can be as simple as highlighting the contents page to include units or pages that you would like the student to complete over the summer keeping those critical concepts in mind.

Another option for summer work can be found in online programs. I have come across three online options for concept practice; *Primary Math Digital*, it’s twin *Math Buddies* and a program new to the US market, *Matholia*.

*Primary Math Digital (Free 15-day trial) *and *Math Buddies (Also a free trial) *are backed by Singapore Math’s Primary Mathematics and Math in Focus series. Both offer students video tutorials that can be viewed by the student (and parent) an unlimited number of times. These videos are scaffolded to follow the pictorial and abstract progression of learning.

Teachers can assign videos, practice and assessment tasks for students to complete over the summer at their own pace. The practice pages can be a little challenging to navigate, but with some initial guidance, students should be able to complete the tasks independently.

Both programs require the school to purchase annual student and/or teacher accounts to gain access to the library of lessons. There are Homeschool accounts available. Expect a price tag of around $30 per student depending on the number of accounts purchased.

Another, more affordable option new to the US market is *Matholia*. *Matholia* was developed by two teachers from Singapore and has been used by teachers and students in Singapore as well as several other countries. This program also includes a library of video tutorials, practice and assessment tasks as well as fact fluency tasks and games.

The videos are easy to understand and are also strategically scaffolded for student understanding. The practice and assessment tasks are intuitive and easy for students to navigate. As with the other programs, teachers can assign tasks for students to complete over the summer.

*Matholia* also requires the school to purchase annual student accounts (teacher accounts are free) but is much more affordable at just $8 per student.

I can’t go without saying that any of these options will give students practice, but struggling students need more than just extra practice working through math problems. They need more time in the concrete phase of learning using manipulatives; base-ten blocks, place value chips, model building with connecting cubes or paper strips, fraction strips or circles, etc. So, please, consider not only sending these students home with books and login IDs but also with a bag of manipulatives for hands-on learning and practice.

Now…back to dreams of lazy mornings and time to relax and recharge. Have a great summer and rest assured that your students will be prepared for the next grade with a little summer math work.

City Springs Elementary/Middle School is a neighborhood charter school operated by the Baltimore Curriculum Project (BCP). We are a conversion charter school, which means we were an already existing Baltimore City Public School that was taken over by an outside operator to bring innovative and research-based curriculum and other programs to enhance the school. To learn more about BCP, click here.

Initially, the school was seeking help with its Middle School math. After I made a pair of on-site visits at the end of the 2014-2015 school year, Dr. Rhonda Richetta, Principal of City Springs, decided to adopt Singapore’s Primary Mathematics Common Core Edition.

I’ve returned to City Springs periodically this year to provide continuing support as the school’s teachers and coaches adopted a Singapore Math® curriculum. The school is making remarkable progress, and I want to share stories written by some of City Springs’ dedicated teachers about the students’ growth during the year.

I’ve clipped an excerpt from each story with teachers’ observations and very valuable insights about the program and why it is working so well for their students. Please click on the links to read complete stories from the school’s website. I love the photos of students showing off their skills and having fun with math!

[Full disclosure: My work assignment at City Springs is contracted through Staff Development for Educators.]As the year winds down and I look back at all that my students have learned this past year, I still feel panicked at what’s left to be covered. This is the end of a 3-year adoption cycle of Primary Mathematics and while I’ve been able to cover more curriculum than in the previous 2 years, I am still left wondering, *“How can I fit it all in?”*

In the fall of 2013, we adopted the Primary Mathematics curriculum in Kindergarten through sixth grades. We knew this would come with its challenges but felt strongly that if we were going to offer our students the “world’s best mathematics curriculum,” then we needed to offer it to all, not just those who made the K, 1, 2 cut.

With this plan, we knew there would be time spent filling in holes in our first year, teaching skills and concepts that the students were missing, and building a solid foundation in number sense and place value. We accepted the fact that we would not cover all of the curriculum that first year, and worked with Cassy Turner to develop a sequence for each grade level that included teaching critical lessons from prior grade levels.

In the second year, teachers were feeling a sense of relief. We’d made it through that challenging first year. We experienced the curriculum from start to finish, well – at least our version of it, and we felt confident. We weren’t faced with the need to back-teach (as much). Our students entered the year having learned and retained a deeper understanding of those critical math concepts.

With Cassy’s advice, we created a new plan for our second year. We knew the lessons that had been skipped the previous year and teachers worked together to map out a Kindergarten through sixth-grade sequence that allowed us to get further through the content, and more importantly, accounted for previously omitted lessons. If we didn’t teach a lesson on geometry to our third graders our first year, we made sure those students would get those lessons in fourth grade our second year.

The year ended, and our standardized test scores showed slight increases in problem solving and algebra readiness, both areas of statistical concern with our previous curriculum.

Entering year three, we felt confident in our abilities to deliver lessons. Along with our students, our staff had developed a deeper, conceptual understanding of math. We were able to effortlessly explain new concepts, differentiate on the fly, and anticipate misconceptions. We incorporated anchor tasks, journaling and finally had a grasp on how to effectively use all of the materials.

We entered the year with the goal of teaching the entire curriculum. Halfway through the year we were teaching material nearly a month ahead of our previous two years and felt really good about it. Then came…

- rehearsals for performances
- grandparents’ day presentations
- spring field trips…
- field day…
- and all sorts of other school commitments.

By April, we had just about lost the scheduling lead that we had enjoyed in December.

So, here I find myself once again faced with the task of choosing one lesson over another and prioritizing the importance of skills and concepts that I may or may not have the time to teach. Fortunately, the list to choose from is smaller than in the years before. I’ve been able to cover almost all of the material, nearly reaching the goal.

I consider myself lucky to have been in a situation of specializing in math in the lower grades over the past 3 years. I have been able to experience the strengths of the sequence, which in my mind, is one of the pillars of success of a Singapore Math curriculum. Going forward, I know it will be easier to thoughtfully prioritize content to eliminate the risk of creating gaps or holes in student learning that could potentially weaken their foundation.

As I leave the school, I look forward to bringing this wealth of knowledge that I gained over the past 3 years to Math Champions and look forward to assisting other schools that are facing the question of, *“How can I fit it all in?”*

Demand for my training and coaching services has grown every year; recently, it has become impossible to accommodate all the schools looking for help with their math instruction.

Expanding has been on my radar for several years, but I did not want growth to affect the quality of professional development services. To take this important step, I needed a candidate who has stellar credentials as a teacher/trainer/coach AND who shares my passion for elementary math.

I’m delighted to share some exciting news. After eight years as a one-person consultancy, I am hiring my first employee.

In a 13-year career, Beth Curran has taught mathematics in Kindergarten through 4th Grade as both a classroom teacher and a Singapore Math® specialist. Beth is currently the Lead Math Teacher and Math Department Chair at St. Anne’s-Belfield School, an independent Pre-K-12 school in Charlottesville, Virginia. At St. Anne’s, she oversaw the successful adoption of Primary Mathematics, guiding a team of teachers and partnering with administrators and parents.

Beth wrote about the St. Anne’s implementation here.

The transition for our faculty to Singapore Math was daunting at first. The unknown can always challenge teachers and schools. Yet, Beth, as a member of a team of newly-minted math specialists, provided key expertise and wise counsel to her colleagues. The result was a palatable sense of relief as the new math vision was manifested daily by strong instruction. She enabled our teachers and school to move confidently in this new direction.

– Fred Chandler, Associate Head of School for Academics, St. Anne’s – Belfield SchoolThere is a depth to Beth Curran’s teaching, and–through her example and through many discussions–she has helped me deepen my own teaching. She prioritizes true mastery of worthy concepts.

– Karen LeMaire, Kindergarten through 4th Grade Math Specialist, St. Anne’s – Belfield School

Beth has provided professional development and ongoing support services to other Virginia schools that have implemented Singapore Math curricula, as well as hosting several informational sessions to schools interested in the Singapore approach. In addition, she has presented at State and regional math educator conferences. Beth is a dedicated, award-winning teacher, presenter, mentor and advocate of making quality mathematics education accessible to all students.

Beth says, “I’m so excited to share all that I’ve learned on my own Singapore Math journey with educators throughout the country, and beyond.”

I’m thrilled that Beth will be joining me in June. Welcome to the team, Beth!

P.S. In addition to hiring Beth, I’ve also formed a new company, Math Champions Professional Development, LLC. More on that development coming soon.

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

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

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!

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!)

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?

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?

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:

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.

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!

Link to event registration pages:

- researchED Maths and Science Conference
- International Forum: Raising the bar! Implications for UK practice in using the Singapore approach to raise standards of teaching and learning in primary mathematics

The National Council of Supervisors of Mathematics is designed for math Lead, Coaches, anyone with a commitment to improving mathematics education. Their vision is built upon the pillars of:

.

First off, **don’t miss my session** where Lauri Susi and I will be sharing the latest in Bar Model/Tape Diagram technology! It’s the last session on Monday, so plan to hang around afterward with questions.

**#1706 A Picture + Technology = Understanding x 10**

Lead Speaker: Cassandra Turner

Co-Presenter: Lauri Susi

Room: OCC 202

Tape Diagrams, Bar Models, and other pictorial representations sit at the intersection of CCSSM, problem solving and technology! Come solve problems from the simple to the complex, and investigate a web-based program and iPad app that will help anyone incorporate this practical and visual problem-solving strategy into their classrooms.

**#1101**** Who Are You and What Do You Want To Create?**

Lead Speaker: Lucy West

What is the most important skill you can cultivate to influence the system in which you work in positive and productive ways? Are you caught in old thinking patterns and outdated notions associated with hierarchical structures and defined roles? Do you have the skill set and the courage to speak up in ways others will listen and even act upon? When the pressures of new standards, high-stakes testing linked to student and teacher evaluations seem relentless and you are faced with tough decisions about where to focus your energy, how do you maintain your integrity yet stay in the game? Join me to reflect on what the research says is the most important condition needed for any organization to thrive.

**#1102**** A Curriculum Developer Looks at the Common Core and Its Testing**

Lead Speaker: Zal Usiskin

The CCSSM constitute an ideal curriculum, while PARCC, SBAC, and the other tests constitute a tested curriculum. To what extent are these curricula in sync with each other and to the NCTM Standards (another ideal), and to what extent do these curricula agree with curricula in other countries?

**#1203**** Digital Tools and Three-Act Tasks: Marriages Made in the Cloud**

Lead Speaker: Arjan Khalsa

Bring your iPad and your inquisitive mind. What do you notice? What do you wonder? How can you lead your district to use free, online tools effectively? This session features mathematical tasks with video anchors and online, virtual manipulatives. Themes will include: inquiry, rich discourse, perseverance, and authentic connections for grades 3–5.

**#1207 Working in Harmony: Orchestrating Effective Parent Education**

Lead Speaker: Barbara Blanke

It is critical to work in harmony with parents to support ALL students’ mathematical learning. Our coaching teams implemented informational meetings, parent coffees, Problems of the Month, and family mathematics nights to nurture parents’ understanding of the CCSSM. Receive ready-to-use resources to educate and build constructive partnerships.

**#1607 Leading Change: Professional Development (PD) Moves That Promote New Ways of Thinking, Learning, and Teaching**

Lead Speaker: Kim Rimbey

Meaningful PD opportunities provide time for participants to develop pedagogical content knowledge while reflecting on practice. But teachers want to walk out the door with “activities they can use tomorrow.” Join us as we examine strategic PD moves that customize and enhance presentations while layering classroom tasks with deep adult learning.

**#3102 Critical Connections**

Lead Speaker: Greg Tang

Teaching students to make sense of problems may be the single most important thing we can do. But what exactly does that look like? Join us as we make surprising connections between part-whole models, comparison word problems and simultaneous equations. Mathematics is amazing when it actually makes sense!

**#3103 Demonstrating Understanding of Algebraic Concepts**

Lead Speaker: Robyn Silbey

Other than following a series of prescribed steps, how can students show they have deep conceptual understanding? First, you’ll explore one algebraic concept that connects ideas and demonstrates true understanding. Then, leader actions and thought processes applying to daily instruction will be shared. Every student can fully understand algebra!

**#3302 How to Create a Mathematics Teacher Specialist Network**

Lead Speaker: Robert Kaplinsky

We have developed a thriving network of over 140 mathematics teacher specialists from five counties that regularly meets to collaborate and save time by pulling the best ideas from the group. Members state that it is the best ongoing professional development they receive. Learn how to grow one in your area and avoid potential implementation issues.

**#3604 Practical Suggestions for Recasting our Homework Policies and Practices**

Lead Speaker: Steve Leinwand

Little in life is a greater waste of time than doing and going over mathematics homework. This session will review typical practices, available research, and propose a set of changes that result in far more impactful homework policies and practices.

The NCTM (National Council of Teachers of Mathematics) Annual Meeting & Exposition program 2016 is now online and it’s time for my annual review of Singapore Mathematics sessions.

I’ve included sessions on Tape Diagrams and Strip Models because there are **no** sessions this year citing Singapore Math® strategies and only one using Number Bonds**.** This is because the Common Core progression documents cite many of Singapore’s visual models. My hope is that many of the visual strategies are modeled throughout the conference. The number of overall choices addressing Singapore strategies is down from the number offered last year.

As usual, most of the sessions overlap. Below are my thoughts on which session to choose, if you have a conflict.

**#23 MMMMM (Making Math More Meaningful with Models) in Pre-K–2**

Lead Speaker: Duane Habecker

Too often we push our students directly to abstract algorithms without first giving students the prerequisite experience with models such as empty number lines, number bonds, arrow method, area model, etc. Teachers will learn how to use these models to make math meaningful for their students.

**#26 Problem Structures for Tape Diagrams**

Lead Speaker: Nirmala Nutakki

Tape diagrams can model relationships in a wide variety of problems involving the arithmetic operations, fractions, ratios, and percent. We will examine the problem structures most amenable to tape diagram solution and illustrate how tape diagrams can be used to develop and support proportional and algebraic reasoning.

**#34 The Front Lines of Modeling: Bar/Tape Models from Real Classrooms**

Lead Speaker: Dr. Kevin Mahoney

Tried your hand at bar modeling? Examine common errors, misunderstandings, and dispositions in actual student work. Leave with a deeper understanding of how children use (and misuse) models

**#34.3 Exhibitor Workshop – Bar Modeling with Math Buddies, the Singapore Math® Online Resource**

Lead Speaker: Marshall Cavendish (Probably Chris Coyne)

Discover how students learn and use the Singapore Math® bar model in kindergarten–grade 5. The foundations set in kindergarten with number sense and number bonds develop into meaningful links in the problem-solving process. Math Buddies, a K–5 digital resource will take your students through the Singapore Math® bar model approach to problem solving.

**Recommendations:**

*Not familiar with Nirmala Nutakki, so can’t comment. Math Buddies is a great digital resource for Singapore Math, and it will also be modeled at their booth. I’m intrigued by the MMMMM, but will probably be in Kevin Mahoney’s session – he wrote his doctoral dissertation on bar modeling in the classroom!*

**#332 Tape Diagrams . . . NOT Just for Early Elementary Grades**

Lead Speaker: Jodelle S. W. Magner

Co-Speaker: Sue McMillen

Participants will examine the use of tape diagrams throughout CCSSM and solve questions from first grade through algebra. Participants will leave with knowledge of where tape diagrams apply in many types of mathematics. Some tape diagram solutions will be compared to traditional solutions to illuminate the usefulness of this tool.

**Recommendations:**

*I’m sure these presenters are wonderful, but I’ll be attending session #323 Insights and Practical Suggestions for Making Coaching for More Effective with Steve Leinwand. If you are a math coach, this one is mandatory.*

**#464 Using Tape Diagrams to Foster Algebraic Thinking and Problem Solving**

Lead Speaker: Bill Jackson

Co-Speaker: Makoto Yoshida

See how tape diagrams can be used to foster algebraic thinking to help young children solve addition and subtraction problems involving unknowns in all positions. Video footage of a lesson study cycle on teaching through problem solving and students sharing and discussing multiple solution methods will be shared.

**#589 Math Talk: Teaching Concepts and Skills through Stories and Illustrations**

Lead Speaker: Char Forsten

A young child’s understanding of the world is enlightened and expanded through stories and illustrations, so it makes sense to use these resources when learning mathematics. Participants will learn how to use “math talk” as a powerful way to provide consolidation and purposeful practice of essential skills and concepts.