Math Coaching at U-32 and Dealing with Apathy for Deep Math

I was back at U-32 this week, which made me feel nostalgic.  Much has changed in middle school math from last year to now: heterogeneous grouping, math on core, a few new babies, a new schedule...  but I am appreciative that our administrators and superstar representatives on the scheduling committee (that's right, CG!) have made it so that the middle level math teachers still have common planning time.  We were able to pull together a coaching schedule that involved planning, modeling, debriefing, data analysis, and more planning. All of this on a (somewhat) half-time coaching schedule!

Focus

The prior week, the middle school teachers and I sat down on at our Thursday math huddle to map out our week of math coaching.  After batting around a few ideas, we decided to focus on seventh grade content and instruction (with our next week of math coaching focusing on eighth grade content).  The content would be operations on fractions starting with multiplication.  The modeling would involve using the "Square 1" concrete area of a rectangle.  The focus of our collaboration would be using formative assessment to inform instruction because we know that this is something we all need to do better.

Each teacher would support me as I introduced the instructional model in one class. Following that, each teacher would teach using the model with my support.  We would analyze student work together to decide where the instruction needed to go next.  

We followed the multiplication of fractions instructional plan that I have mentioned in previous posts.  Since this is a fifth grade non-negotiable skill, we were hoping to move quickly, more quickly than I had been able to move in the elementary classes to date, but it was not to be.  In all three classes, we saw that students could generally (but not universally) multiply fractions procedurally, but they could neither model it nor contextualize it.

This was our dreamy schedule:

Monday
Before TA, Hollis and I planned a bit.

Band 1: I taught one of Hollis' seventh grade class with his and Katie's support.

Band 2: Hollis taught his second class and I supported.

Band 3: Hollis, Katie and I were able to meet for 20 minutes look at student work and talk about what to do next.  We developed some resources for the next step, moving to a conjecture.   

Tuesday
Katie and I were able to "chat" and collaborate online about this benchmark assessment end of seventh grade.  She got feedback from the other math folks at their Tuesday meeting.


Wednesday

Katie's "Favorite No."  This student almost has it, but is confused by the 
"non-routine" scale.  This showed us that they need to see scale as something that 
varies.  We built Katie's next interventions around it.

Band 1: Katie and I looked at her last exit card for her Math Skills class and planned her next interventions and exit card.  We decided that students weren't accessing the fraction work because they were having difficulty with varying scales and interpreting pictorial representations (see example at right).

We also looked at her 8th grade Math Strategies class.  She asked me how to take the work she had been doing with them on fractions with Cuisenaire Rods to the next level.  We worked on that together.


Band 2: I taught Scott's class with support from Scott, Katie, and Heather (who took copious notes).  As and exit card, we asked students to fold "square 1" to model two fourths times three fourths labeling: the dimensions and area.

Band 3: I modeled using Cuisenaire Rods as a concrete model for variable expressions and equations (and we threw in substituting fractions for variables ad evaluating those expressions, too).

Lunch: Katie, Scott and I debriefed and looked at his exit cards.  We found some mistakes that could provoke good discussion.

This is what we were hoping to see. Two fourths times three fourths represented as the area of a rectangular "Square 1." 

This one is mistaking "two halves" for "two fourths" in the horizontal length

This one is confusing area with side length.

Katie's Math Skills Concrete-Pictoral-Abstract
lesson on representing operations on a number line.

Band 4: I was able to connect with Hollis to touch base with him about how his students were doing with arriving at a conjecture, proof and application.  Katie taught the lesson we worked on in the morning in Math Skills.  I dropped in to observe.

In addition to the confusion over scale, when we looked at the student work on fractions, we noticed that some of Katie's students were not sure how to interpret magnitude on a linear representation.  Should they be counting points or the lengths between the points?

I knew her plan, but she surprised me by working in negative integers, too.  It was great.  Students created concrete representations (measuring using the rod to measure length and end of a rod to mark a "point."  Then she had them represent operations concretely, recording it pictorially on the number line and using a symbolic expression.

Thursday
During our Middle Level Math Huddle, we all looked at Scott's exit cards as a group.   We sorted them by evidence of reasoning and talked about what understandings and misconceptions students demonstrated, and how to proceed.

We revised our lesson somewhat for Cathy's class.

At Scott's request, I modeled the conjecture and proof piece, as well as the model for multiplying mixed numbers.  Here's a video that details that:

Multiplication of mixed numbers is a good opportunity to "test" the students' initial conjecture: Does it still hold up?  This is also another great opportunity to revisit the term "rational number?"  What is a rational number?  Is a mixed number a rational number?  How do you know?  Etc.

Friday
Cathy talked to me about the gaps in fractional understanding that she noticed when students were working on area models for compound probability.

Band 1:  I taught Cathy's class with her support (and Linda helping and taking notes).

Band 2:  Cathy taught her second class with my support.

Band 3: Cathy and I looked at her exit cards and debriefed.  We decided that she will continue to do "warm ups" and practice problems next week (relating multiplication of fractions to probability) and tie up some loose ends.  After vacation she will focus on the conjecture, area proof and application, application, application.

This is what we spent ALL of our time on at Doty a few weeks back.  So this post would be good to revisit if you're looking for resources on that.  Speaking of resources...

Resources

We developed these new resources this week:

• A lesson plan that is grounded in their recent work on finding probabilities of compound events.
• This tiered formative assessment on Multiplicative Operations on Fractions was our warm up.  Students work for 5 minutes independently to fill in a row of their choice.  We used this information to see what kind of review of the area of a rectangle model was needed.
• An exit card - students were given a fresh "Square 1" and were asked to model a multiplication problem, labeling the dimensions and area.
• Homework on Multiplicative Operations on Fractions (other than the last example involving a negative number this would work for 5th or 6th grade, too). 
• An organizer to help facilitate moving from the exploring phase to coming up with a conjecture and area proof.
• Katie Jarvis sneaked this video of me working with an 8th grade intervention group on variable expressions using Cuisenaire Rods and posted it on youtube.

Question of the week

"Why are we still multiplying fractions?  I already know this!"

Here we arrive at the apathy portion of this post...  I have fielded this question (and others like it) many, many times (even from my own fifth grader at the dinner table).  And I know that I am not alone in this.  The procedure for multiplying fractions is one of the easiest to memorize. Even students who struggle with nearly every other procedure seem to be able to remember this one.  Indeed, our data showed that nearly all of them knew the procedure.  So, why were we spending precious seventh grade math classes going back to "Square 1?"

Thanks to Cindy Gauthier (who teaches 5th and 6th grade math at Berlin) for lending me her copy of Knowing and Teaching Elementary Mathematics, by Liping Ma because she has given me the perfect language to address this question.  In her seminal book comparing U.S. and Chinese math instructional practices, Ma emphasized the need to shift from a math reasoning system that is based on "how" to one that also demands "why?"  We need to practice this as teachers so that we can ask it of our students.  "How" is important, but isn't worth a scratch without "why."  We get at the "why" by asking students to model and communicate the reasoning behind the procedure, the principles involved, and by asking them to apply it to routine and non-routine contexts.

When I was teaching multiplication of fractions last year to my seventh graders, I preempted this question (luckily, it had already come up when we were multiplying and dividing whole numbers again) by initiating a discussion on Growth Mindset and then presenting them with a copy of our Levels of Proficiency for operations on numbers.  I told them that in order to "know" operations on fractions they needed to be able to:

• Perform operations on fractions accurately and with understanding using the efficient algorithms and methods.
• Model all operations on fractions using discrete models, linear models, and the area of a rectangle model.
• Explain the reasoning behind the methods using multiple representations (including concrete materials, pictorial symbolic models) as opposed to simply memorizing a process.
• Explore and explain why for positive numbers, multiplication of a factor by a fraction less than one results in a product that is smaller than the factor and division by a fraction less than one results is a quotient that is larger than the dividend;
• Generate equivalent fractions by multiplying (or dividing) the numerator and denominator by a common factor and can explain why the product is an equivalent fraction using the identity property of multiplication (any factor multiplied by one or a fraction that is equivalent to one has the same value as the original factor).
• Apply all operations on fractions to multiple contexts - both in mathematics and the real world - and be able to contextualize and de-contextualize problems involving all operations on fractions.

I gave my students some examples to think about, and they realized three things: 1) They didn't know as much as they thought they did. 2) Deep understanding of fractions makes it easier to learn about proportionality and algebra. 3) I was going to be holding them accountable for deep understanding.

Interestingly, we found that some of our folks who are usually "top" students had a difficult time creating the physical models and became frustrated.  This is a great time to remind these students (who are unaccustomed to struggling) about Growth Mindset.  It is okay to struggle and work through it.  

This is also a great time to remind them of the Math Practices we've focused on so far this year and how they all apply:

To sum up:  focusing on "why" isn't easy, but it's essential.  And we can use our definition of proficiency and the Math Practices to help get us there.


That's all for now, math nerds.  As always, let me know if you have any requests.  Next week, I'm off to EMES again to work with the 5th grade teachers, and I'll try to get more resources together for dividing fractions.