Thursday, December 26, 2013

Why and How do we Share Student Work?

It is now easier than ever before to share student work.  With a simple click of a button we can capture what a student is creating in our classrooms and make it available for the world to see!  And why wouldn’t we take advantage of this remarkable opportunity to share a student’s justification, thought process and/or evolution?  Our featured article demonstrates The Power of Student Work

So maybe I've convinced you just how impactful this process can be.  But the next question is how do we authentically go about sharing student work with our colleagues?  The Massachusetts Department of Elementary and Secondary Education has partnered with the Center for Collaborative Education to promote and support two cohorts of professional learning communities (PLCs) in twenty-six districts in Massachusetts.  One of the key features of a PLC is the ability to follow a protocol to structure discussions and maintain focus.  Check out this relevant Learning From Student Work-Protocol.  It articulates guidelines in selecting appropriate student work and helps structure reflection and dialogue around the selected pieces.

Last but not least, I’ve included some actual examples for the visual learners in our blogging community.  Expeditionary Learning has collected hundreds of samples of student work available for the public to view and discuss. The student work can be selected by discipline and grade level.  My personal favorite (featured at the top of this blog post) is the The Home of Ivy Olcott.  This piece explores standards 5.MD.1 and 6.RP.3 through an interdisciplinary math and art project.

Wednesday, December 18, 2013

Teaching Algebra to 8th Graders

There is a major debate taking place right now amongst math educators: is it possible to offer an authentic Algebra I course to 8th graders while still ensuring that all 2011 8th grade math standards get taught?  Teaching Algebra I to 8th graders used to be a no-brainer.  There seemed to be much repetition within the math standards in the middle grades, and Algebra I was offered to ‘advanced’ students who were ready to move more quickly.  But now that Massachusetts has fully adopted the 2011 Curriculum Frameworks for Mathematics, the new Model Algebra I course is incredibly rigorous and forces topics like simultaneous linear equations, functions and exponents down to the 8th grade.  The following crosswalks and teacher resources cautiously offer a pathway of teaching Algebra I to 8th graders while ensuring that no key standards get omitted along the way.

The document below offers a pathway to taking Algebra I in the 8th grade by condensing 7th grade, 8th grade and Algebra I standards into the 7th and 8th grade years.  Go straight to page 80 to see the Compacted Middle School Pathway.  The document shows exactly where standards get condensed and even offers a potential breakdown of units per grade level.

If our ultimate goal is to get students to take AP Calculus by senior year of high school, then there is also the option of taking enhanced math classes in grades 9th, 10th and 11th grade.  In the Enhanced High School Pathway to Calculus the department offers a way to take Algebra I for the first time in 9th grade but still work towards AP Calculus in 12th grade. 

Methuen Public Schools has done a tremendous amount of work around creating an accelerated math program starting in the 7th grade.  Julie Ward, Methuen's Math Coordinator, has included a few resources in her dropbox.  These documents have proven critical in notifying parents, selecting students and crosswalking standards.

A word of caution: this debate has in no way been resolved.  I am not necessarily approving any of these cited pathways, but simply showing you different approaches to this tall task.  It must be said that teacher knowledge of the Model Algebra I course and student readiness play an absolutely integral role in this discussion.

Wednesday, December 11, 2013

What is PISA and Where do we Stand?

PISA helps us compare our students with other students across the globe.  Students are asked to demonstrate knowledge of reading, mathematics and science literacy on the assessment.  There were a total of 65 participating countries and education systems and this was Massachusetts' first time administering the assessment.

Want to take a look at some of the world-wide results?  The presentation below includes additional background information, sample questions, and aggregated data in the form of bar graphs and scatterplots.

Findings From the 2012 PISA Report

If you are more interested in the Commonwealth's results, the following two links will provide you with Massachusetts' specific data.  The first is a press release from the Department of Elementary and Secondary Education (DESE) and provides bulleted data points including lists of the top performers.  The second is an opinion piece from US News and World Report that argues that the Bay State's success has come from setting the expectation of high standards.

DESE's PISA Press Release

US News and World Report Highlights MA Using PISA Data

Wednesday, December 4, 2013

More Conceptual, Less Procedural!

The exposé below uncovers commonly used tricks in the math classroom.  These tricks are exposed as a way to teach simple memorization as opposed to deep conceptual understanding.  The seven chapters are organized by math domain for easy readability, but the index also breaks down the tricks by specific math standards.  We have all taught mathematics by introducing tricks and pneumonic devices, but this exposé will change the way you expect students to demonstrate understanding!

A Guide to Avoiding Shortcuts That Cut Out Math Concept Development

Looking for real world math resources that help teachers step away from the procedural?  Look no further.  This article highlights four incredibly useful online tools that provide relevant mathematical videos, activities and lessons.

4 Tools to Connect Students to Real World Math

Wednesday, November 27, 2013

A Timeline for PARCC Implementation

The Massachusetts Department of Elementary and Secondary Education (DESE) has announced a clear PARCC timeline! After two years of field testing, the state Commissioner will recommend whether to adopt the PARCC assessment as the state testing program.  As you can see in the timeline below, a final decision will be made in late fall of 2015.  The recommendation to slow down and reflect a bit more about Massachusetts' adoption timeline is a direct response to districts' recommendations. 

Feel free to look at the DESE Website for more information about the revised PARCC plan.

2013–2014 School Year


November 19, 2013:

Board votes on plan for two-year transition from MCAS to PARCC



Spring 2014:


PARCC field test administered in randomly selected Massachusetts schools/classrooms (and in 13 other states)

2014–2015 School Year



Early fall 2014:

Analysis of PARCC field-test data by Department staff

Late fall 2014:

Board receives update on PARCC field test and operational test that includes an assessment of whether PARCC is on track to be better than MCAS
Winter/spring 2014–2015:
Massachusetts schools administer first operational PARCC or MCAS assessments, pending Board approval of plan for districts to choose between the two; all grade 10 students will continue to take MCAS in order to qualify for the Competency Determination

2015–2016 School Year



Summer/early fall 2015:

Standard-setting for PARCC tests; analysis of operational data

Late fall 2015:

Board votes on full adoption of PARCC based on a determination of it being better than MCAS


Spring 2016:

Administration of PARCC operational tests for grades 3-8; grade 10 students will continue to take MCAS to qualify for the Competency Determination

For an excellent graphic that connects and embeds all of the PARCC initiatives, click on the link below:

Making Sense of PARCC

Thursday, November 21, 2013

'Teaching' the Mathematical Practices

The Standards for Mathematical Practices are the only math standards that span all the way from Pre-K through 12th grade.  The hope is that even adults in the workplace engage in these practices and procedures!  Although we expect that both kindergarteners and precalculus students reason abstractly, what the practices actually look like in the classroom range tremendously when it comes to both age and ability.  This week we are focusing on ways that mathematicians can engage with mathematics using the practices and how this might be explicitly taught and formatively assessed.

Looking for objectives that match with specific Mathematical Practices?  Look no further.  A group of Wisconsin educators created a two page document that summarizes the eight Mathematical Practices into kid friendly objectives.  Although the practices are skills that are developed over time, it is possible to breakdown these characteristics into smaller, more attainable goals. 

The practices can be extremely difficult to assess.  This matrix was developed by Leadership, Coaching and Mathematics (LCM) as a rubric that can place students into the initial, intermediate or advanced categories when it comes to the Mathematical Practices.  They have also broken down commonly used teaching strategies and lined them up with the Mathematical Practices.  This is a nice way to view what a teacher may already be doing in the classroom and match those strategies with the development of specific practices.

EDC is offering a course that will feature tools, resources and instructional routines that are designed to support students with learning disabilities to develop and use the Mathematical Practices. This course runs from February 26th-28th and focuses specifically on grades 4-8.  Participants will receive 2 graduate level credits and walk away with practical tools to use in the classroom.


Thursday, November 14, 2013

Can Everyone Learn Math?

What do you say to a student, teacher, administrator or parent when they assert that specific students simply aren't capable of learning in the math classroom?  Why is it that math has the reputation of being a subject that students either understand or don't? Are the common core standards going to make students' math anxiety worse?  What are some strategies that teachers can use to help boost math confidence?  The three highlighted articles below address all of these ubiquitous questions and more.

'Is it True That Some People Just Can't do Math?'
Daniel T. Willingham is a psychologist at the University of Virginia.  In this editorial, Willingham uses science to prove that all students are fully capable of learning K-12 mathematics.

'The Stereotypes About Math That Hold Americans Back'
Jo Boaler is a passionate professor at Stanford's Graduate School of Education.  In this article, she clarifies how the math common core standards can improve students' confidence in the math classroom.

'The Biggest Lie Students Tell Me (and How to Turn It Around)'
José Vilson is an incredibly genuine writer who currently works as a math teacher and coach.  José offers four key strategies that teachers can use when students claim that they can't complete a task.

Tuesday, November 5, 2013

Technology in the Common Core Era

The Transition to Online Testing

Both PARCC and Smarter Balance are currently incorporating technology-enhanced questions within their online assessments.  This promises to completely change the way we test our students in the immediate future.  When is this transition being fully implemented?  Will the format of the online tests be different than the traditional pencil and paper approach?  Will we get to try out this new assessment tool prior to full implementation?  This article from Education Week answers all of these questions and more.  A must read for 21st Century teachers, coaches and administrators!

Online Practice

We now know that online exams are in our future, but we do not know how to prepare for this major shift. Looking for ways to practice using some online tools?  Assessments-The Next Generation has broken down the computer-based practice into grades 3-5, grades 6-8 and grades 9-12.  The website also includes sample assessment items as well as interactive resource sites.  All links are free but many require Flash or Java to operate.  Take some time to play around with these rich tools.

Monday, October 28, 2013

Conceptual Understanding of Fractions

Fractions are a major focus of the Common Core Standards for Mathematics.  Although many of us were not taught fractions in a conceptual or visual way, we are now expected to teach beyond the procedural; and for good reason!  Below are some useful videos, tools and explanations that might help facilitate this shift in the math classroom. 
Using a Number Line to Teach Fractions
This video does an excellent job showing and explaining how the open number line (and double number line) can be used when teaching fractions.  It demonstrates how students can model with mathematics (standard for mathematical practice #4) and even includes a few examples of student work.

Fractions Progression
This website highlights seven key units of fractions and the ideal progression of these units.  It connects 3rd, 4th and 5th grade fractural concepts together through videos and sample teacher explanations.

Visual Fractions Practice
This tool is a visual learner's dream!  It includes fraction visualization models, worksheets, games and step-by-step explanations.  Everything on this site is free and kid-friendly.

Monday, October 21, 2013

Rigorous and Open-Ended Tasks

It's hard to keep all of the assessment terminology straight these days. 

What is the difference between a CEPA and a performance task?  How do we know when we've created an authentic formative assessment?  What's the point of administering interim assessments?  How do we define rigor?

This week's blog focuses on rigorous tasks and gives you concrete examples to work with.  Regardless of what we name these tasks, we know that it is imperative that we increase the amount of open-ended, rigorous problems that we present in the math classroom.

If you've explored the Partnership for Assessment of Readiness for College and Careers' (PARCC) website, you know that these tasks are a lot more open-ended than they used to be.  Below are some websites that provide you with sample mini-assessments or performance tasks.  I hope you find these resources useful and that you try out a few!

Achieve The Core
Achieve The Core-Mini Assessments

The Mathematics Common Core Toolbox
Elementary, Middle and High School Sample Tasks

Middle School Tasks
Open-Ended Math Problems

Friday, October 11, 2013

Academic Vocabulary in the Mathematics Classroom

Academic Mathematics Vocabulary Website

This website has highlighted math vocabulary lists that are based on the Common Core State Math Standards.  The lists range from kindergarten to high school.  You do not need to create a login to see the 'Math Vocabulary Words at a Glance' at the bottom of the webpage.


Building a Bridge to Academic Vocabulary in Mathematics

This article gives many practical tools that educators can use to help build academic vocabulary in the math classroom.  Beyond introducing purposeful activities strategies and investigations, this article stresses the importance of educators using formal mathematical vocabulary on a consistent basis.


Wednesday, October 2, 2013

Good News for Spanish Speaking Students, Parents and Teachers

PARCC's Math Tests to Be Translated Into Spanish

The common math assessments under development by PARCC will be translated into Spanish and possibly other languages as well.  Massachusetts is one of eight states in the consortium to say that it needs a Spanish-language math assessment.

Smarter Balanced Assessment Consortium Launches Spanish Webpage

Smarter Balanced has launched a Spanish webpage with factsheets, practice tests and videos for teachers, parents, and students. The webpage provides information on the assessment system and the Consortium’s work to meet the needs of English language learners.

Tuesday, September 24, 2013

Common Core Updates

Welcome back to the Greater Boston DSAC mathematics blog!

My name is Leah Tuckman and I am the new DSAC Math Specialist for Greater Boston.  The following three links are intended to keep you updated on recent Common Core news.  Feel free to leave comments or questions.   

1.  34 Common Core Aligned Model Curriculum units from MA are now available on ESE website

You must sign up to receive access to these units.  All you need is an email address!  Here is a breakdown of the mathematics units available:  one Pre-K, one 2nd Grade, two 4th Grade, one 6th, one 7th, two 8th, and one Algebra 1 unit available. 

2.  "11 Tips on Teaching Common Core Critical Vocabulary"

Academic vocabulary has been directly linked to the Common Core in every subject.  If you are looking for a list of commonly used and tested roots, prefixes and suffixes to help deconstruct academic vocabulary, click here.

3.  "Common Core Assessments Release Calculator Guidelines"

Calculator usage is becoming much more uniform across the United States.  Take a look at this article to get a better sense of what PARCC and Smarter Balance are using as guidelines.

Tuesday, August 27, 2013

PARCC Prototypes

Dear Colleagues:

I want to share with you some task prototypes and samples developed for the Partnership for Assessment of Readiness for College and Careers (PARCC).  They are designed in accordance with the Common Core State Standards, and include examples for English Language Arts and Mathematics.

In addition to this, the Common Core Map for Math Graph is now available in pop up text! 
You may download it at:

Or, get the PDF version at:

Enjoy your holiday weekend.

Thursday, August 22, 2013

Greetings, Greater Boston Math Blog readers!  Greater Boston DSAC is writing to share with you a video selection from The Teaching Channel entitled, "Discover Number Patterns with Skip Counting."  It follows a lesson on choral counting by 200's, allowing students to identify multiple patterns and structure.  The instruction is differentiated through multi-colored visuals, lecture (require active listening), students working collaboratively and reporting back to their fellows and the instructor.  The lesson is built around math practices from the Common Core, encouraging students to construct viable arguments, exercise critical thinking, and justify conclusions.  In the end, all main points are reiterated to reinforce student learning.


Monday, June 3, 2013

Webinar Series - Archived from ASCD - Classroom Instruction that Works

Heading into the end of the academic year educators often take time to reflect on the year and plan for the next! I recommend this collection of four archived webinars from ASCD on "Classroom Instruction the Works" to help spark some thinking. Watching these in your PLC or with peers would make for some nice informal PD.

Tuesday, May 28, 2013

PARCC - incorporating UDL principals

Worth a read is Katie Novak's recent blog entitled "Even the PARCC Does UDL" -

"Standardized assessments are finally catching up with UDL! The PARCC Draft Accommodations Manual was just released, and although it is “pre-decisional” (p.2), it includes “universally designed” (p.5) supports that will help all students demonstrate their knowledge and skills on the assessment."

The intended accommodations and technology scaffolds embedded in design of the test fulfill two of the guiding principals of UDL "multiple means of representation" and "multiple means of presentation".

You can download a copy of the Draft Accommodations Manual here.

Wednesday, May 22, 2013

Mathematical Practice Standards...connections to ELA and Science practices

As I have written previously, the cornerstone(s) of the new frameworks for mathematics are the math practice standards. Phil Daro, common core author has said “If the Standards for Mathematical Practice are not in place, well then, you are not really using the Common Core”.  

In the spirit of all teachers, not just math teachers, becoming aware of these practices I would recommend using the graphic and summary below to spark interdisciplinary conversations. It is based on work by Tina Chuek from and was recently posted by National Science Teachers Association.

Monday, May 13, 2013

3 Rules to Spark Learning...

 last week included many exceptional talk about education  All eight can been seen here: I am sharing one that especially resonated with me for mathematics and science teaching. 

Ramsey Musallam's three (simple) rules stem from his notion that "student questions are the seeds of real learning".
  • #1 Curiosity comes first.
  • #2 Embrace the mess.
  • #3 Practice reflection.

Here is his presentation:

Wednesday, May 8, 2013

ESE Opportunity - Director for Mathematics, Science and Technology Engineering

Current Opening: Director for Mathematics, Science and Technology Engineering 

The Office for Mathematics, Science, and Technology/Engineering (MSTE) at the Department of Elementary & Secondary Education (ESE) is responsible for writing and updating curriculum frameworks in mathematics, science, and technology/engineering. MSTE manages related initiatives in curriculum and instruction, such as advanced placement programs, professional development, content institutes, mathematics and science partnerships, liaison networks, the financial literacy pilot program, and advisory councils.

The director for MSTE is responsible for supporting districts and educators across the state with programs and resources to help them improve student achievement in math, science, technology, and engineering education. The incumbent will manage and lead the development of these initiatives in accordance with state and federal requirements governing planning for and delivery of services, and will lead the planning, management and development of statewide policy for academic leadership, curriculum development, program and staff supervision, staff deployment, and fiscal management for MSTE programs.

The director will also be responsible for managing the day-to-day operations of the office that includes leading, supervising, coordinating and overseeing the work of staff, and the overall budget/spending planning and resource allocation for the unit.

Monday, May 6, 2013

NCTM Denver Meeting

If you were unable to attend the 2013 annual  meeting of the National Council for Teachers of Mathematics here are two ways to capture the meeting contents

1. From the NCTM web site
2. From the MathRecap site

Monday, April 29, 2013

"Tiny Math Games"

This week I am borrowing from an excellent blog dy/dan, Dan Meyer.

Dan's following is quite substantial and the stream of replies to his posting about "tiny math games" is an terrific resource for a mathematics classroom. I added a few of my own - look for the Norma Gordon comments on the blog page.

Here was his prompt for tiny math game suggestions:
  • The point of the game should be concise and intuitive. You can summarize the point of these games in a few seconds or a couple of sentences. It may be complicated to continue playing the game or to win it, but it isn't hard to start.
  • They require few materials. That's part and parcel of being "tiny." These games don't require a laptop or iPhone.
  • They're social, or at least they're better when people play together.
  • They offer quick, useful feedback. With the multiplication game, you know you don't have the highest product because someone else hollers out one that's higher than yours. With Fizz-Buzz, your fellow players give you feedback when you blow it.
  • They benefit from repetition. You may access some kind of mathematical insight on individual turns but you access even greater insight over the course of the game. With Fizz-Buzz, for instance, players might count five turns and then say "Buzz," but over time they may realize that you'll always say "Buzz" on numbers that end in 5 or 0. That extra understanding (what we could call the "strategy" of these tiny math games) is important.
  • The math should only be incidental to the larger, more fun purpose of the game. I think this may be setting the bar higher than we need to, but Jason Dyer points out that people play Fizz-Buzz as a drinking game. [Jason Dyer]

Tuesday, April 23, 2013

Curriculum Alignment and Mapping Webinars from MA ESE

The Massachusetts Department of Elementary and Secondary Education, in partnership with the Pioneer Valley Readiness Center, the Lower Pioneer Valley Educational Collaborative, and the Collaborative for Educational Services, will present a series of free webinars on curriculum alignment and mapping during the month of May.  

 To register, please go to the links below:

May 13, History and Social Science: 10-11 (elementary) and 1-2 (secondary), History and Social Science 

May 21, English Language Arts and Literacy: 10-11 (elementary) and 1-2 (secondary)

May 29, Mathematics: 10-11 (elementary) and 1-2 (secondary)

Monday, April 22, 2013

Creating a Rigorous Task and Making Sample Tasks Your Own

This week I am sharing a resource I created for a workshop focused on identifying language objectives together with mathematical content and practice standards objectives. It is based on the Buttons Task (grade 6 Statistics and Probability) found at (illustrative Mathematics is an initiative of the Institute for Mathematics & Education funded by the Bill & Melinda Gates Foundation)

This is an example of how a task found on-line (or elsewhere) can be modified for broader learning objectives and rigor while embedding [student directions] to encourage mathematical discourse.

If you are interested in a copy of the file email your request to:

Content Standard:
Develop understanding of statistical variability. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
Context for language use:
The Jar of Buttons task requires READING the definitions and questions, WRITING, recording reasoning and questions. Additionally there is well SPEAKING AND LISTENING with the oral discourse activities such as turn and talk.

A Jar Full of Buttons
Zeke likes to collect buttons and he keeps them in a jar.
Occasionally Zeke empties the buttons out of the jar so
he can see all of his buttons at once.

PART ONE: Review in your group the following definition:
A statistical question is one that anticipates an answer based on data that vary. 

Example: “How many minutes do 6th grade students typically spend on homework each week?” is a statistical question. We expect 6th grade students’ time spent on homework would vary from student to student or week to week. 

Non-example: “How much time did Juana spend studying last night?” is not a statistical question since the answer is based on a single number individually complete the table below. 

PART TWO: Which of the questions in the table below are statistical questions? Explain why it is or is not a statistical question.


Statistical Q?
Yes or No
Why or Why Not?
 I. What is the typical number of holes for the buttons in the jar?

 II. How many buttons are in the jar?

 III. How large is the largest button in the jar?

 IV. If Zeke grabbed a handful of buttons, what are the chances that all of the buttons in his hand are round?

V.  What is a typical size for the buttons in the jar?

 VI. How many buttons have four holes?

VII. How are these buttons distributed according to color?

PART THREE:  When everyone has recorded his or her answers, compare with your partner/group. Share reasoning if there are any disagreements. 

Individually write TWO new “button questions”. One should be a statistical questions and the other one not statistical. Be prepared to share your thinking and explain why it is or is not a statistical question.

PART FOUR:  Trade your two questions with your partner/in your group and see if they agree with your classifications.  OR Hand in your two Question as exit ticket 

Statistical questions I, III, IV and VII
Possible statistical question:  What is the distribution of button shapes in the jar?
Possible non-statistical question:  How many more buttons are in the jar this week?

Adapted from Illustrative Mathematics