November's Math Practice of the Month

Reason Abstractly and Quantitatively

This math practice is an essential focus if we are to get all of our students to the deepest level of mathematical proficiency, the Application and Communication level (level 4).

What's the standard? 

Mathematically proficient students: 

• Make sense of quantities and their relationships in problem situations. 
• Decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols.
• Contextualize—to make meaning of quantities and symbols in terms of a situation. 
• Create multiple coherent representations of the problem attending to the meaning of quantities and units involved, not just how to compute them; and 
• Know and flexibly use different properties of operations and objects.

What does it look like and how do we teach it? 

You can click here for a video and lesson from Inside Mathematics that focuses on representing numerical patterning.

We've also got some great examples of this happening right now in WCSU.  Lisa Hanna, who teaches 6th grade at Doty Memorial School, had students create these poster-sized graphic organizers to demonstrate (and develop!) their fluency between context and abstract representations of ratio problems: 

At U-32, 8th grade math classes are using a "4 Representations of a Function" graphic organizer to develop links between abstract and contextual representations.

Cathy Guiffre (who is out on leave from teaching 7th and 8th grade right now because she is home with her new baby boy!) takes decontextualization one step farther with her "waffle" problems.  Here's Cathy expressing her disbelief that I've never heard of "waffle" problems before:

No, they don't involve breakfast food.  Apparently, "waffle" is the British term for "extraneous information."  Cathy takes a word problem, adds in some extra bits of information and prints all of the bits up on separate cards like this:

Then, she collects the cards into shuffled bundles and hands them out to her students.  Her students' first task is to decide which of the bits of information matter.  Then, they decide how to decontextualize the problem and represent it symbolically.  I should add that this would work at many different grade levels, and students find it very engaging.

I hope this has given you some ideas for how to integrate this math practice into your practice.  As always, if you have questions or something you are burning to share please comment below or send me an email at edorsey@u32.org.

I can't wait to see your ideas!

First Quarter Math Coaching

It was a very active first quarter in our fifth through eighth grade math classrooms with teachers embracing the opportunity to collaborate and stretch and grow in new ways.  We are all learning together, and I have been amazed by how much we have been able to accomplish together in such a short time.  I want to take this opportunity to highlight some of the work that we have been doing.

At the in-service on October 10, I was able to work for half the day with the entire fifth and sixth grade team and half the day with the seventh and eight grade team.  Information about the workshops can be found here.   One of our tasks for the workshop was to identify goals for our WCSU math teams.  We identified these goals: 

• We will work collaboratively to develop clear and consistent language, instructional models, scope and sequencing, and definitions of proficiency.  
• We will develop and implement common assessments that will allow us to respond to needs and assess the effectiveness of interventions.  
• We will improve access by developing and using coherent concrete models for each non-negotiable skill and employing strategies to move all students from the concrete models to abstract models with conceptual understanding. 

As a starting point, our focus for this year is to make sure that every student in grades 5 through 8 is able to use the area of a rectangle model for multiplication and division reasoning for all rational numbers.  If mastered, the area of a rectangle model provides a conceptual thread that can bring students from operations on whole numbers in third grade to operations on all rational numbers in seventh grade to operations on polynomials in high school.  I spent time during the October 10 inservice introducing the model and how it can be used to teach for conceptual understanding and procedural fluency. 

Here the area of a rectangle model is being used to teach the distributive property of multiplication in eighth grade Algebra.

Since September, I have been in every classroom in grades 5 through 8 modeling instructional methods, co-teaching and supporting teachers as they learn to use and apply the area of a rectangle model and language.

The next few weeks will be very busy, too, with members of the WCSU math steering committee developing common assessments with Karin Hess, a day of embedded professional development with Mahesh Sharma and the launch of our new screening tool, easyCBM. 

   

September's Math Practice of the Month

Make sense of problems and persevere in solving them

Click here for a student friendly poster explaining what this practice means.

This math practice is a great focus for the beginning of the year because it implies that we create a classroom culture where struggle is okay, where students take the time to analyze and explain rather than rushing to a solution and where making mistakes and rethinking the process is all part of the pathway to greater learning.  This can provide a solid foundation for all of our math instruction.

What's the standard? 

Mathematically proficient students: 

• Start by explaining to themselves the meaning of a problem and looking for entry points to its solution. 
• Analyze givens, constraints, relationships, and goals. 
• Make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. 
• Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. 
• Monitor and evaluate their progress and change course if necessary.
• Use multiple representations (concrete objects, diagrams, equations, verbal descriptions, tables, graphs, etc.) to help conceptualize and solve a problem.
• Check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" 
• Understand the approaches of others to solving complex problems and identify correspondences between different approaches.

What does it look like and how do we teach it? 

Here is a video from The Teaching Channel showing how one 3rd grade teacher approaches teaching this math practice.

Click here if you can't see the video.

There are additional videos available on the Inside Mathematics website that you can check out if you have the time and inclination.

Additionally, I have posted some graphic organizers (including the one used in the video) on the WCSU Math Tools and Templates page of our district's math website.

As I visit classes this month, I will be looking for ways that we are addressing and assessing this practice to share out with you all.  If you have something you are burning to share please comment below or send me an email at edorsey@u32.org.

I can't wait to see your ideas!

Celebrating what's right...

Jason Reichert

Jason's Questioning Wall

Jason Reichert teaches high school mathematics (mostly Algebra) in room 217 at U-32. On the wall above the document camera at the front of his room, he hangs posters of prompts to ask students who are presenting their work.

Jason knows that it's important that students have rich dialogues about their mathematical thinking and problem solving approaches.  The questioning wall prompts are visual reminders to Jason and his students to ask divergent questions that clarify and deepen understanding. 

Collaborate with Jason 

Jason is all about improving on his practice though collaboration with colleagues.  When I first noticed Jason's questioning wall last year and commented that I liked it, he immediately asked me if I had any ideas for how he could improve or add to the questions to make them better.  So I know he will appreciate your comments and ideas about adapting or improving upon his good idea.  Can you help Jason?  

Please share your ideas with us by adding a comment below!

  

Questions for the WCSU Math Coach

Who is the Math Coach?

My name is Ellen Dorsey, and I am very excited to be supporting teachers this year as the Math Coach for Washington Central Supervisory Union!  I taught math in middle school and high school at U-32 before becoming the Math Coach.  Additionally, I have worked with teachers from the elementary grade levels through college on proficiency-based differentiated instruction strategies.  

I live with my husband and two school-aged children in Waterbury, Vermont where I serve on the local school board.  This Fall, I start a doctoral program in Educational Policy and Leadership at the University of Vermont, which should fit nicely with the half-time Math Coach job.  Outside of school, you can find me playing in the woods in running shoes or on skis, reading or making something with my hands.
  

What does the Math Coach do?

It is my job to provide classroom coaching and modeling of instruction and assessment practices, to provide training for teachers and to consult with teachers on curriculum, assessment and instruction.  The focus for this year is on supporting teachers in grades 5 through 8, but I am happy to discuss comments, questions and suggestions and to help problem solving around math instruction in WCSU at all levels.

Collaboration is what I have enjoyed most about teaching, and I believe that connecting teachers in meaningful and productive ways is the best catalyst for professional growth that will benefit us (as teachers) and our students.  As we embrace the challenge of improving the performance of all students in our district, we will need to work together to develop and highlight models for best practices.  I see the Math Coach as central to facilitating and supporting that collaborative work.

How do I contact the Math Coach?

It is very important to me to be available to teachers.  I have a desk in the WCSU Central Office, but I will be reaching out to teachers and will spend much of my time working "on-site" with teachers.  I have created this blog to share news and highlight best practices in mathematics across our district.  I also created a Washington Central Supervisory Union Math website that contains resources and information.  I will be keeping my short term action plan and schedule updated on that site as well.  

Otherwise, please email me at edorsey@u32.org, and I will get back to you.  I'm quite obsessive about checking email. 

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