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.