I spent this past week in San Antonio attending the annual conferences of the National Council of Supervisors of Mathematics and the National Council of Teachers of Mathematics as I do most years, and as always, I had a fantastic time and learned a ton.

Just a quick overview of some of the highlights:

** Learning, Planning, and Teaching Together: Facilitating Job-Embedded Professional Development**, Kendra Lomax and Becca Lewis (University of Washington). I feel incredibly strongly about practice-based PD, and since I also happened to know that Kendra got rave reviews from a group of K-2 teachers she'd recently been doing some practice-based/job-embedded PD with, I was excited to hear what she and her colleague had to say on the topic. We heard about how they've been exploring the idea of rehearsal as a way of teachers preparing to teach a lesson, where the rehearsing involves very focused and thoughtful real-time coaching from a teacher educator. They shared some interesting videos, and I look forward to exploring more of their work on on the TEDD (Teacher Education by Design) website.

** Using Identity and Agency to Frame Access and Equity**, NCTM President Elect Robert Q. Berry III. Some colleagues and I recently presented a couple of seminars with this year's WestEd/SVMI Network cohort on the topics of Mathematical Agency and Authority as well as Access and Equity; it's an incredibly important topic, and one around which I feel I am continually deepening my own understanding. I appreciated Dr. Berry sharing some excellent concrete examples of how it's possible to leverage what students bring with them to the classroom to

** Coaching Teachers on the Topics of Fractions, Ratios, and Rates**, Vanessa Cerrahoglu & Jody Guarino (Illustrative Mathematics). Part of my interested in upper elementary mathematics has to do with how those ideas progress towards key middle school topics like proportional reasoning and rates of change. In this session, the facilitators asked us to think carefully about fractions, ratios, and rates, particularly how they are the same/related and how they are different. I actually think that a lot of teachers are often a bit unclear on some of those final points (I recently attended a scorer training where a number of high school teachers argued vehemently that the ratio 1:3 has the same meaning as the fraction 1/3), and the structure they had us engage in felt like a great way to get kids (or PD participants) thinking carefully about those finer points without feeling "dumb" or defensive. It also seemed easily adaptable to other topics where you might want to get students or PD participants talking about at fine distinctions between related ideas.

*Routines for Reasoning: Ensuring All Students Are Mathematical Thinkers;*** Using Routine Rehearsals to Transform Teaching Practices;**, and

**, Grace Kelemanik, Amy Lucenta, & Claire Nuchtern (Fostering Math Practices). Our WestEd/SVMI team is hard at work putting together our 2017 Summer Institute, and a couple of months back we decided to use Grace and Amy (along with co-author Susan Janssen Creighton)'s fantastic book**

*Meeting the Needs of ALL Students through Instructional Routines**Routines for Reasoning: Fostering the Mathematical Practices in All Students*as one of our primary resources. We are always looking for ways that our teachers can support their students with the Standards for Mathematical Practice, and the instructional routines outlined in the book are very accessible and approachable for a wide range of practitioners. Since we're still fleshing out our Summer Institute, there was no way I was going to pass up a chance to hear the authors talk more about the ideas presented in the book and their implications. I enjoyed seeing a number of routines modeled in person (Calculate and Contemplate, ...). I also loved all the slides and additional resources available on their site, www.fosteringmathpractices.com.

** Expanding mathematical knowledge as number domains change from whole to rational: An example in the context of division** and

**, Deborah Schifter, Susan Jo Russell, Reva Kasman, and Virginia Bastable. What with my ongoing quest to deepen my understanding of upper elementary mathematics, how to teach it, and how to coach it, I'm always on the lookout for interesting sessions on topics like fractions and division as well as what working on the Standards for Mathematical Practice can look like in those grade levels. In the first session I attended with Virginia, she took the group through an abbreviated version of a PD activity that invited teachers to think about division problems like 32 divided by 5 and what the answer might be (6 2/5? 6.40? 6? 7? 6 OR 7?) depending on why we're dividing 32 by 5. (Thinking about 5 divided by 8 was even more interesting!)**

*Mathematical Argument in the Elementary Classroom*In the second session I attended with all four, they discussed their recent work on representation-based arguments, i.e., getting elementary students to use concrete models (diagrams, manipulatives, story situations, number lines, etc.) to start think about how they can generalize properties of operations. For example, students might look at pairs of equations like 5 + 8 = 13 and 6 + 8 = 14, and 22 + 7 = 29 and 25 + 7 = 32 and work on describing and articulating the pattern they see. Eventually the teacher has the students work on trying to generalize the pattern using some concrete representation like snap cubes or drawings. Helping younger students begin to move from the concrete and specific towards generalizations is a tough job, so I look forward to sharing these ideas with some of the elementary teachers we work with at WestEd. Also can't wait to check out their new book, *But Why Does It Work? Mathematical Argument in the Elementary Classroom*.

** 6 x 2/3 or 2/3 x 6: Using Structure and Precision to Build Understanding of Fraction Multiplication**, Ryan Casey (Boston Public Schools). Yet more fun with fractions! In this session, Casey pushed the group to think more deeply about multiplying rational numbers -- ie, problems that look like A x B where A and B are both rational. Each factor can be a whole number, a unit fraction, a non-unit fraction, an improper fraction, or a mixed number, giving us 25 different "types" of problems! But developmentally, how should we proceed through these problem types with students? It may seem intuitively that the multiplication problems 20 x 1/4 and 1/4 x 20 are the same, but in terms of how students think about and make sense of them, they're very different. Casey went on to illustrate how students can use mathematical structure and the properties of operations to make sense of different types of rational multiplication problems, then build on their understanding to conceptually tackle more and more sophisticated problem types, all without being formally shown a single algorithm.

** Fractions: Too Important to Teach ½ Way**, Lynne Nielsen (Louisiana Tech). But seriously, I cannot get enough about fractions and division. Dr. Nielsen had us all work a fraction division problem, then selected a variety of solution strategies to share, organizing them from most concrete to most abstract and illustrating the natural progression along which most students' understanding of the topic proceeds. She also guided the group towards a conceptual answer of the age old question, "But why invert and multiply?" She also shared a resource she's found to be particularly useful when coaching pre-service teachers on this topic,

*Extending Children's Mathematics: Fractions & Decimals*, by Susan B. Empson and Linda Levi.

My colleague Katie Salguero and I presented at NCTM on Friday, which I'll post more about sometime soon!

**Did you attend NCSM or NCTM this year? What were your favorite sessions?**

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