Model Drawing (Bar Modeling or Singapore Math)

September 9, 2014 2 comments

Last year I was introduced to a strategy known as Singapore Math.  I have since seen it referred to as Bar Modeling or Model Drawing.  I was hesitant about any “new” strategy and very reluctant about anything that would take away from the beauty of “pure mathematics”.  It only took one short training session to convince me that this is what had been missing in my classroom.  I have tried many techniques such as tables, charts, and graphic organizers to help students with word problems with little success.  This is one strategy that spans grade levels and abilities to help all students be successful in math.  There are many great, free resources to help teachers get started.

I met with a fifth grade team today to work through a story problem that appeared very difficult at first:

Meg has 120 flowers at her flower stand.  Of all the flowers, 1/4 are red,

1/8 are pink, and 1/8 are blue.  Of the other half, 1/3 are yellow and

2/3 are purple.   How many are there of each color?

If you are like most students, a problem with this much information and this many fractions scares you.  I could see the the wheels turning in their heads as they were thinking  things like:  common denominator, multiplication of fractions, can we skip this one?

We first wrote the answer with a blank for our numbers.  I love this technique for so many reasons!

Meg has __ red, __ pink, __ blue, __ yellow, and __ purple flowers.

We then decided what we were counting or talking about:  Flowers.  We drew a unit bar after the word flowers.  (Hint:  If the problem  involves fractions, use a long unit bar.  You will probably have to divide it.)

Model Drawing One



Next, we read the problem and pause to add in the information:

“Meg has 120 flowers at her flower stand.”

Model Drawing Two


“Of all the flowers, 1/4 are red.”  This means I need to divide my drawing into fourths.

Model Drawing Three


“1/8 are pink, and 1/8 are blue.”  I need to go back and divide the other pieces are divided into eighths.

Model Drawing Four


“Of the other half, 1/3 are yellow.”  Oh no, I should not have divided the second half of the bar into eighths.  I will create another bar underneath it and divide it into thirds.

Model Drawing Five


“and 2/3 are purple.”  How many of each color?

Model Drawing Six



Looking at the first bar, there are 8 sections, so each section contains (120 divided by 8) 15 flowers.

Model Drawing Seven


The second bar represents half of the 120 flowers, there are 60 flowers to share among three groups.  This means there are 20 flower in each subgroup.

Model Drawing Eight


Now, all a student has to do is count of the number of flowers for each color group.

Meg has _30_ red, _15_ pink, _15_ blue, _20_ yellow, and _40_ purple flowers.

The great thing about the bar modeling process is it can be adapted for all levels of mathematics.  I have seen fifth grade students solve a systems of equations using the bar modeling method.  I know that all students do not need to use this method, but it is important that even the accelerated learners have a method or tool to use when the math becomes challenging.

Some great sites to visit for resources include and


Categories: Algebra 1, Common Core

Back in the Classroom

August 28, 2014 Leave a comment

The teachers in our district have embraced the Integrated Mathematics, but we still have a lot of questions about the task based learning approach of Mathematics Vision Project.  I love the curriculum, but we are changing drastically how we teach students and that takes time.  I volunteered to return to the classroom to teach a few lessons so I could experience the new curriculum.  It is one thing to sit in my office and imagine how great it is and another to stand in front of a group of 16 year old students asking them to create and use mathematics.  Here are a few things I learned from my day back in the classroom:

1.  Teaching is exhausting.  My voice was gone by the end of the day and my feet were tired.   I was both physically and emotionally exhausted at the end of the day.  I forgot this part of teaching.

2.  I love students.  Kids are fun and will say and do the most outrageous things.  I work mostly with adults in my job, so I forget how  refreshing and innocent 9th grade students can be.  (And maybe not so innocent.)

3.  Task based learning requires a different method of planning.  I used to rely on my math skills to make it through a lesson.  I could create a problem off the top of my head and demonstrate it for a group of students.  When we are working a task and a student tries a different approach, I need to know if that pathway is valid and where it will lead.  There is no glossing over a task.  I need to know that task and be prepared for anything.  In other words, I need to work the task myself.  I can honestly say I did not work all the demonstration problems myself when I used to teach.

4.  I need community.  I used to teach with my door shut and in isolation.  I planned my lessons by myself.  I was happy to share what I created online or with a teacher down the hall.  I was also more than happy to steal from other teachers, but I never had conversations with others about what actually happened in my classroom.  This week when I taught Math 1 and Math 2, I invited a group of educators to sit in the back of the room.  It was great to talk with them afterwards to receive feedback.  If we want to improve as educators, we must be open to letting others observe and provide feedback.

5.  I work with some great teachers!  It was nice to see the routines other teachers established in their classrooms and to reap the benefits of teaching a class where the students knew the expectations.  It was so nice to hear the teachers excited about the new curriculum.  One teacher told me that his students, who typically struggled with math, were understanding deep concepts and were able to teach others after the task based approach.  Our teachers have experienced so much change in the last few years and I admire their ability to adapt.

My new plan for my job is to make sure that I step back into the classroom on a regular basis.  I don’t want to forget the joys and challenges that a classroom teacher faces.  I don’t want to create or implement theory that doesn’t translate to practice.  Plus, I really love working with kids.

Mathematics Vision Project: Task Based Learning at its Best

July 1, 2014 10 comments

Recently our district has decided to pursue the Integrated Mathematics approach for high school math.  I am really excited about this transition.  Due to the lack of curriculum resources, we decided to use an online, free curriculum.  The Mathematics Vision Project is an integrated curriculum created by educators from Utah through a grant from the state.  I have looked at the curriculum online and love it, but it wasn’t until I went through their two day training that I realized it is truly the best math curriculum I have ever encountered.  Here is why I love their program:

1.  Task based learning focused on the learning cycle.

Task based learning is the rage right now with Common Core.  Our state, Tennessee, put teachers through intensive training on how to teach math using tasks.  The problem with this is teachers were left to create their own tasks or find random tasks.  In order for task based learning to work, the tasks must be sequenced appropriately and build on the previous learning.  MVP does this.  They have different types of tasks for different purposes and they are sequenced to build on each other.  The learning cycle involves Developing Understanding, Solidifying Understanding, and Practicing Understanding.  When you look at the tasks in a unit, the tasks are labeled as one of these.  This helps for both students and teachers to understand the purpose of the task.  Some tasks only develop the understanding.  Later, only after a teacher can guide a class discussion, are students expected to apply and practice the new learning.  This idea of different types of tasks for different stages of learning is critical.


2.  The have low threshold and high ceilings.

I was amazed with the multiple entry points for the tasks.  It felt as if any level of student could do something.  Often with tasks though, the mathematics is “dumbed down.”   This is not true for MVP.  The tasks are rich and have high ceilings.  If you have a group of student who finish early, there is always something in the task to stretch the learning.

3.  Story contexts throughout the module.

Take a look at Module 2 in Math 1.  It starts with a rich task about two children starting a pet sitting business.  The purpose of this first task is to start students down the pathway of thinking of multiple constraints on a variable (systems of equations).  Students will use this context throughout the entire module adding a little more information with each task.  Students should feel as if they are invested in a Problem Based Learning approach, broken into small, obtainable chunks.

4.  Not just what to teach, but how to teach it.

Most curriculum contain what a teacher should teach, but little about the best methods for teaching.  This is the first curriculum I have encountered that explicitly helps the teacher know how to teach the standards.   Each problem or exercise has a purpose:

  • Teach new knowledge
  • Bring misconceptions to the surface
  • Build skill of fluency
  • Engage students in Math Practices

5.  Meaning full homework and practice.

Practice is done by experts… Doctors practice medicine and Lawyers practice law.  Why would we send home practice when our students have not mastered the material?  This creates frustration and with Common Core, it leads to parents posting crazy math homework on Facebook.  MVP has amazing, thought out homework assignments.  They divide the homework into three categories:

  • Ready:  Things a student needs to review to be ready for upcoming work.
  • Set:  Things we did today in class that you need to practice to solidify understanding.
  • Go:  Things students should be “good to go on.”  This is review material.

Each assignment also has links to online videos to help review concepts students may not remember.  (I know in reality, that my students may not have done the homework, but I could use this as starters and exit tickets in my class.)

6.  Flexible Curriculum

Since the MVP curriculum is online, it can be updated at any time.  This means if something isn’t working or their are mistakes, they can easily be fixed.  This is not true of traditional text books.  The MVP team did hint that they are currently working to align the tasks and material to release them in a traditional math pathway.  This means that if your district does not do Integrated Math, you will still be able to use the MVP curriculum.

Overall, MVP offers a great curriculum and fantastic professional development.  I encourage you to attend an event and at the least, take some time to review the material.


NCSM Reflection

April 11, 2014 1 comment

I recently attended the National Council of Supervisors of Mathematics Conference in New Orleans.  I was honored to receive the Iris Carl Grant.  This grant pays for your travel expenses to attend.  If you have not attended a NCSM conference in the last three years, you ought to consider applying.  It was an invaluable experience.  I wrote a reflection of the conference to submit to NCSM.  I am including it below.

46th Annual NCSM Conference, New Orleans, LA

Reflection by Amber Caldwell, Recipient of the 2014 Iris Carl Grant

K-12 Mathematics Coordinator, Bradley County Schools, Cleveland, TN

All of my classroom experience did not prepare me to serve in the role as K-12 Mathematics Coordinator for my district.  After fourteen years in the classroom, I thought I was equipped and had the skills to serve the teachers in my district.  After a few months in this newly created role, I realized that enacting change in my classroom or at the school level was easier than trying to inspire an entire district consisting of 16 schools.  I enrolled in an online class with Jo Boaler offered free though Stanford University.  I was introduced to concepts and ideas that amazed and humbled me.  I realized after the course, that I needed more.  I am grateful to NCSM and the Iris Carl Travel grant for allowing me the opportunity to attend what I hope will be the first of many NCSM Conferences.

Mike Schmoker, in the opening session, reminded me that as educators and leaders, that less is often more and we need to focus on the basics of a great lesson.  He stressed the importance of clarifying, practicing, and mastering first things.  The most effective strategies that a teacher and school can implement are curriculum, cold calling, and 90-120 minutes of purposeful reading and writing.  This session reminded me to encourage and maintain the basics while trying to implement the CCSSM.

Cynthia Callard, Jane LaVoie, and Stephanie Martin offered a session on using the Progressions of Common Core State Standards to Deepen Teachers’ Content Knowledge.  While I have read the Progressions and used them while developing curriculum maps, I had not considered using them in a professional development setting with teachers.  One of my goals in my new position is to create Professional Development modules to be utilized in Professional Learning Communities to introduce units of study to teachers and initiate the planning stage.  Through this process, I would like to help teachers solidify and deepen their content knowledge.  After this session, I will incorporate the Progressions into this planning.

I had the honor of hearing Cathy Seeley speak and was very excited to receive her new book upon arriving at NCSM.  She reminded the attendees that all math students need to understand, do, and use mathematics.  The understanding is making sense of mathematics, while the doing consists of facts, skills, and procedures.  When a student uses mathematics, they are modeling, reasoning, and thinking.  The mathematical habits of mind require students to perform thought experiments.  While the habits of mind were not new to me, the reminder to focus on them was much needed.

One of the highlights of my time in New Orleans consisted of hearing Marilyn Burns speak on Linking Formative Assessments to the CCSSM.  While I had been exposed to the Mathematics Reasoning Inventory site before, to have her explain the process and anecdotes around building this project was invaluable.  I am excited about bringing this resource back to the teachers in my district.  I see how important it is to incorporate Math Talk into our daily routines and to ask students to explain their reasoning.  According to Marilyn Burns, students who lack understanding of a topic may rely on procedures too heavily.  True understanding comes from explaining and critiquing the reasoning of others.

The moment that inspired me the most was hearing Jo Boaler speak on Erasing Math Inequality.  Dr. Boaler’s work in the field of mathematics education is motivational and encouraging.  She presented five barriers to high and equitable math achievement.  I am sad to admit that I and my district are guilty of creating some of these barriers for children, and one of my missions is to remove these obstacle so all students can achieve.  Dr. Boaler encouraged us to change our beliefs regarding students and their ability to do mathematics.  She stressed that we are harming our students by placing them in ability groups and creating a fixed mindset.    I also see the need to encourage our teachers to look beyond one dimensional mathematics and to teach math and not calculation.  We can do this by encouraging sense making in mathematics.  I was convicted to try and remove timed mathematics testing from our district.  Dr. Boaler’s research shows that timed tests create math anxiety at an early age.  Math should not be associated with speed.  Jo Boaler speaks with such conviction while offering encouragement.  I left her session with a clear mission, determination, and a desire to enact change.

The opportunities for networking and collaborating were invaluable.  Being outside of the classroom and in a leadership role can be isolating at times.  I am so thankful to NCSM and the Iris Carl Travel Grant for this opportunity to attend such a valuable conference.  I am returning to my district renewed and impassioned to be a catalyst of change and a resource to the teachers and students whom I serve.


Categories: Common Core, General

Major Work of the Grade and Vertical Alignment

March 17, 2014 5 comments

I recently stumbled across a great website that offers a wealth of professional development on PARCC and the CCSS, both in ELA and Math.  The site is powered by the National Math and Science Initiative.  Teachers and administrators will need to create a free login to access the site.  The site gives the following information as to why it was created:

“The PARCC Educator Leader Cadres (ELCs) will help each state build and expand the number of educators who understand, support and feel ownership of the successful implementation of the Common Core State Standards (CCSS) and PARCC assessments.”

I found an activity on the site focusing on the Critical Areas in Mathematics.  I printed out the cards on colored paper and asked the Instructional Coaches to place each critical area into the correct grade level.  The cards are below:

Critical areas game K-5 A blue cards

Critical areas game 6-8 A and B green cards

Critical areas integrated A 9-11 pink cards

Critical areas traditional 9-11 A yellow cards

photo1 (1)

After teachers worked together, I gave them this page (2.2 Critical Area Activity Sheet ) and asked them to check their answers and record any discrepancies.  It is important to note that this activity calls the topics the Critical Areas and this differs slightly from the “Major Work of the Grade” as laid out by PARCC.  PARCC’s use of Major Work of the Grade is more specific and the Critical Areas activity creates fewer and broader categories that do incorporate the major work of the grade.

It is a great activity for teachers to see the vertical progression of topics in mathematics with the CCSS.  The site went a step further and created this great one page document ( 2.3 Summary Critical areas summary) over viewing the critical areas from kindergarten to high school mathematics.  The site also included a one page overview of the fluency standards (Key Fluency Expectations Recommendations and Examples of Culminating Standards).  I encourage coaches to print these pages out and laminate them for every teacher.

The site offers great activities on math practices, examining coherence in one domain, and text complexity.  I strongly encourage coaches, administrators, and department chairs to visit the site and utilize some of the great resources.

Categories: Common Core, PARCC Tags: , ,

CCSS Math Resources

February 26, 2014 Leave a comment

It seems as though everywhere you look you can find Common Core math resources.  This is both a blessing and a curse.  I remember years ago (before Common Core) trying to search the internet for resources on math topics such as imaginary numbers or adding rational expressions.  I think that is why blogging drew me in and I started this blog.  I was having a hard time finding great tasks, and using vetted tasks from teachers like me was a blessing.  I wanted to share!

But now we have the CCSS and that means there should be a lot of great resources out there.  Be careful!  I have come across a lot of resources labeled CCSS and PARCC, only to find weak content and revamped activities.  Not everything has to be new, but everything should be aligned.  I wanted to share one of my new favorite sites for finding great resources for CCSS in mathematics.  It has become my ‘one stop shop’.


Here is why I love this site:

  • This site take the best resources and organizes them in one location.
  • It is easy to search by standards to find tasks.
  • You will find links to all the curriculum maps released by states.
  • There is no password or login required!!

Here is part of the mission statement from the site:

“There is so much that has been created by so many and it is out there free to the public via the internet. However, it remains difficult to sift through it all to find the best things for our children to use. This site will hopefully allow teachers to spend more time teaching and give kids more of an opportunity to learn both at school and at home.”

You will find resources from

  • Khan Academy
  • Learn Zillion
  • Mathematics Assessment Project (MAP)
  • NCTM Illuminations
  • Science Net
  • Texas Instruments
  • Dan Meyer
  • Hot Math
  • and many more…
Categories: Common Core, PARCC Tags: , ,

Common Core & PARCC Sample Items

February 18, 2014 1 comment

I have been working on planning professional development for my district on Common Core State Standards and the PARCC assessment.  There is so much information on the PARCC website and it can be overwhelming for a teacher to navigate it.  My goal is to try and weed through the information and present only what is necessary and beneficial to teachers.

PARCC recently released sample questions in their intended environment.  This means the computer-based tools such as drag-and-drop, multiple select, text highlighting, and an equation builder are all active.  It is a great opportunity for teachers to see what computer skills are necessary and how students will navigate the assessment.  This sample assessment does not reflect a complete PARCC assessment.  The questions on the online assessment are all previously released sample items. The one frustration that I have is that  the questions are separated by grade bands and not grade levels.  In my experience, teachers want to focus on their grade level, although I think it is important to be aware of what comes before your course and where students are heading.  To help teachers and administrators, I have created the following documents to support teachers while they are looking at the online PARCC environment.  The documents address each questions content standard(s), grade level (course), and math practice.  Detailed scoring guides and explanations of the questions can be found on the PARCC website under the respective grade band.  Please feel free to provide feedback in the comments.

PARCC Computer Based Samples Grades 3-5

PARCC Computer Based Samples Grades 6-8

PARCC Computer Based Samples High School




Why My Son Needs Common Core

February 3, 2014 7 comments

I know that there is a debate regarding the “new” math common core state standards.  I understand parents are frustrated with children having to learn “new” ways to add, subtract, multiply, or divide.  I understand that parents are frustrated with children having to show work and defend an answer, even when the answer is correct.  I understand your frustration as a parent.  I have a child who is off the charts in math.  He consistently scores in the 90th percentile and above on all standardized assessments.   He just “gets” math.  (His mom is a math educator.)  He is always frustrated when I ask him to defend his answer.  His typical response is, “because I know it’s right.”  I used to think common core was not written for children like him.  He does not need to draw a picture or learn a “new” way to divide.  I was wrong!  My son needs common core.

A few weeks ago we sat down and worked several TCAP (Tennesse State Assessment) type problems for math homework.  They were all division problems similar to the one below:

John has 12 apples.  He wants to share them with 3 friends.  How many apples does each person receive if John gets the same amount as all his friends?

My son was flying through these problems.  After a few moments of watching him, I realized he wasn’t even reading them.  I stopped him and asked him what he was doing.  This was his explanation:

“Mom, the lesson is on problems with division.  I just divide.  The bigger number always comes first, so I take the bigger number divided by the smaller number.”

Something inside my math teacher heart died.  I wanted to scream, “The bigger number doesn’t always come first!” and “What if the problem was multiplication and you assumed wrong?” and then I realized that our curriculum and check list standards have reduced real life mathematics to this.

A week later my son’s need for Common Core became evident.  We were at Publix grocery shopping and we came to the juice aisle.  Orange juice was on sale, 3 for $6.00.  At Publix, you do not have to buy all 3 to receive the sale price.  My son started to put three juice cartons in the cart.  I stopped him and explained we only had to buy one.  I then asked him, “If they are on sale for 3 for $6.00, how much is one carton of juice?”  Remember, my son was in the 98th percentile last year in math and he “gets” it.  His response, “$2.50? $3.00?”  What?!  We stopped in the grocery store and got out paper and pencil and I made him show me how he arrived at his answer.  He drew a picture.  Through this process, he realized his mistake.  He told me he didn’t realize it was a division problem.  He said, “Mom, I know 6 divided by 3 is 2, but I didn’t realize this was a division problem.”  So yes, my third grade son sometimes needs to draw pictures.  Memorizing his math facts is not enough.  He needs to understand the situations that necessitate the memorized facts.  He needs to be taught strategies to solve problems when they seem unfamiliar.  He needs Common Core.

Categories: Common Core, General

Why I Blog… for @k8nowak

December 3, 2013 1 comment

I took some time last week to scroll through my own blog and could not believe some of my old posts.  I hardly recognized my own words.  I am definitely not the same educator or the same person I was when I started this blog over three years ago.  I have changed.  My job has changed.  Education has changed.  Kate Nowak played such an integral role in starting me on the path to blogging so I wanted to respond to her request:

1. What hooked you on reading the blogs? Was it a particular post or person? Was it an initiative by the nice MTBoS folks? A colleague in your building got you into it? Desperation?

Dan Meyer.  I found Dan Meyer one summer through desperation.  My principal asked me to serve as department chair and I felt so unworthy.  I had a 25% failure rate in my classroom.  I had students who hated my class and did not see the purpose in being there.  From Dan’s blog I found Kate’s and Sam’s.  All of these blogs showed a passion for teaching I had never seen or experienced before.  I wanted to be a part.

2. What keeps you coming back? What’s the biggest thing you get out of reading and/or commenting?

Teaching is hard.  It is rewarding, but hard.  I see teachers everyday struggle and cry.  I see teachers leave their rooms in joy with the desire to share their successes and I see them hang their heads and want to hide from their failures.  I see how overwhelmed some of them are with all of the changes Common Core is bringing.  It is more necessary now to build a free community of resources and support for teachers.  We can not and should not do this alone.  If what I write or say can help even one teacher, then it is worth my time.

3. If you write, why do you write? What’s the biggest thing you get out of it?

I started writing as a window into my classroom.  I wanted to share what worked and what failed.  I now work at the district level and have access to hundreds of classrooms.  This is a huge responsibility and honor.  I feel like writing about these experiences gives me the opportunity to share to a larger audience.  I write to push myself.  Right now, I found myself going off on a tangent (I deleted it) and started writing an I wish I would have when I was in the class room list…  That will be a later blog post.  This just goes to show that blog writing forces me to reflect and push myself to improve.  It really is a selfish exercise.

4. If you chose to enter a room where I was going to talk about blogging for an hour (or however long you could stand it), what would you hope to be hearing from me? MTBoS cheerleading and/or tourism? How-to’s? Stories?

I would love to hear the nuts and bolts of how to start a blog.  Also, how to handle reading blogs and not get overwhelmed.  I remember a time when I wanted to just shut down because I could never get caught up with my blog reading or what I wanted to write about.  Baby steps…  Oh yeah, and twitter.  Twitter and blogging go hand in hand.

Categories: General

Active Vocabulary Walls

November 20, 2013 1 comment

I recently visited several schools in Florida to observe Spring Board in action.  I was impressed with the curriculum, but beyond that I was impressed with the level of engagement of the students.  Students and teachers utilized some great strategies in the classroom.  I was also very impressed with the consistency among the schools across the district.  Every classroom I visited had an active vocabulary wall.  I have never been successful with vocabulary walls.  I would try and hang words on the wall, but I never did anything else with them.  This changed my mind.

IMG_1348 IMG_1362 IMG_1364 IMG_1374

(These cards are attached to a ribbon and have the definition on one side and word on the other)

All teacher are required to have academic vocabulary (Tier 3) posted on the wall.  Every teacher has a unique twist to their word wall.  The definition is always included and students are encouraged to get up and use the wall during class if they need to refresh their memory.  One teacher even moved words to the “Mastered” wall after the class demonstrated a thorough knowledge and understanding of the word.  The teachers would cycle the words according to the unit of study.  I absolutely loved that the words had the definitions hidden, but accessible to students.  If I was still in the classroom, I would find space for this on my wall.


Categories: Common Core