We started Day 2 with Andrea leading a discussion of the burning questions from the day before, such as “How might ideas from Day 1, like understanding mathematical concepts as adjectives or nouns, play out in our classroom?” We then moved to a review of Day 1’s math homework. Teachers found a partner within their grade band (K-2, 3-4, or 5-8) to compare and contrast their solutions to the Film Problem. When we started the whole-group debriefing of the math homework, Fran asked the participants to show her, with thumbs up or thumbs down, if they were satisfied with their understandings of #1 and #2 based on their small group work. Seeing that everyone was satisfied, Fran stated the she was not going to do a whole-group review of the solution to the problems (no need since everyone was good with their solution). Fran then asked if everyone was good with a solution to #3 and saw that there was some hesitation. A number of teachers shared their solutions to #3 (using different representations: tables of values, equations/expressions, graph). Fran then asked people who were not satisfied with their solutions to #4 and #5 to work further on those problems, using the new knowledge that they had built during this session.

Teachers then gathered in grade band groups and completed posters based on these prompts:

- What ideas about expressions, equations, and multiple representations were involved with the hw problem?
- How do similar ideas play out in your grade band?

We then moved onto small-group discussions of the pedagogy homework. Teachers shared curriculum materials in their groups and discussed the math tasks in their curriculum through a cognitive demand lens. Fran and Andrea ended this session with some whole-group discussion of cognitive demand, sharing information about the QUASAR project and how cognitive demand might be a useful lens to use with textbook adoption groups.

We spent the rest of the morning discussion how counters can inform our understanding of numbers and addition. We used geometry to gain a new view of even and odd numbers in terms of stacks of 2, and extended those ideas to looking at remainders when numbers are divided by numbers other than 2.

After lunch we tackled the “100 days of Christmas” problem, which led to a lot of fruitful discussions. There is a summary of many of the arguments made in class (and others) at http://betterexplained.com/articles/techniques-for-adding-the-numbers-1-to-100/.

We next began our discussion of subtraction by basing it on the missing-addend form of addition. Nola shared how students at the end of first grade are already seeing addition with missing addends, which sets up subtraction. We then moved into multiple models of subtraction, both in terms of counters and a number line, eventually settling on distance as a view that will be most flexible.

We also began to explore what the adjective-noun theme has to say about subtraction. The short version is that it works the same as addition, which isn’t surprising considering the deep connection between subtract and addition.. We also considered alternate algorithms for multi-digit subtraction as a test of our understanding of how place value interacts with subtraction.

### Math Homework

“Karl Friedrich Gauss Library Overdue Fees” (Treat the variation as a challenge)

“Problems on Subtraction” (pp. 3.15-3.17) Problem 7

### Pedagogical Homework

Read *Principles to Action*: Section titled “Build Procedural Fluency from Conceptual Understanding” (p. 42-48).

Also read: Webb, Bozwinkel, & Decker. *Beneath the Tip of the Iceberg: **Using Representations to Support Student Understanding.* MTMS, 2008. (Handout from class today)

In your notebook:

- Identify a procedure or skill that you consider essential for students at your grade level to learn. Describe the conceptual understandings that support students’ learning of the procedure or skill.
- Write questions you have based on the 2 readings.

### Odds and Ends

The group that wrote the Common Core Standards are also writing documents detailing how certain ideas progress across grade levels. These are currently in draft form, but are usable. Drafts are available at http://ime.math.arizona.edu/progressions/. A table summarizing the various combinations of given and missing terms in an addition or subtraction problem appears on page 7 of the K-5 Progression on Counting and Cardinality and Operations and Algebraic Thinking.

Andrew uses the following to help him research origins of terms and notation

- The Online Etymological Dictionary. etymonline.com A searchable general-purpose etymology dictionary. My wife and I visit this site at least twice a week.
- Earliest Known Uses of Some of the Words of Mathematics: http://jeff560.tripod.com/mathword.html
- Earliest uses of Various Mathematical Symbols: http://jeff560.tripod.com/mathsym.html