Fran began the morning by sharing a number of resources:

1. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School by Thomas P. Carpenter, Megan Loef Franke, and Linda Levi. (www.heinemann.com)
2. Classroom Discussions: Using Math Talk to Help Students Learn by Suzanne H. Chapin, Catherine O’Connor, and Nancy Canavan Anderson. (mathsolutions.com)
3. Extending Children’s Mathematics: Fractions and Decimals by Susan B. Empson and Linda Levi. (www.heinemann.com)
4. Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades by Susan Jo Russell, Deborah Schifter, and Virginia Bastable. (www.heinemann.com)
5. Putting the Practices into Action: Implementing the Common Core Standards for Mathematical Practice K-8 by Susan O’Connell and John SanGiovanni. (www.heinemann.com).

We went deep into the homework on processes, comparing answers and pulling out some solid lessons on how to invert multi-step processes.   Along the way we also established the convention that a number-based process is most generally written by using a single-letter variable as the initial input. Jen also made the connection between finding inverses and fact families: effectively we have a two-member fact family  when we write a process and its inverse.

After the break we finished the discussion of the rice-and-water problem and how inverses play out there.  The original problem asked “Given R cups of rice, how many (W) cups of water do we need?”  The inverse problem asked “Given W cups of water, how many (R) cups of rice can that make?”  We ultimately came up with formulas for converting between R and W in terms of multiplying by a rate (the exact number we multiplied was 5/3 or 3/5 depending on whether we started knowing R or W).  By drawing both relations on the same axes (with different sets of labels), we drew an important geometric connection between a proportional relationship and its inverse.  Specifically, the graph of the inverse formula looks like the reflection across the line y=x (which can also be seen as the 45-degree line when two axes have the same scale, also the line through (1,1), (2,2), (3,3), etc).

The remainder of the math discussions  (morning and afternoon) centered around the problems in Unit FA-3 on Currency Conversion, both with and without an exchange rate.  This provides a vivid contrast between proportions (without exchange rate) and the more general idea of a linear relationship (with exchange rate.

The pedagogy discussions centered around “focusing versus funneling” question styles, and using student knowledge to guide your process.  After a discussion of what these terms mean, participants broke into pairs and engaged in role-playing exercises.  Partners recorded themselves as they took turns acting as students (who made errors) and their teacher.  The homework is to listen to one’s turn as teacher and analyze one’s teaching style.

Math Homework

Complete all 12 problems under the “Currency Conversion Problem with Exchange Rate”

Read pages FA3.9 to 10, on the terminology of linear relationships.

Pedagogy Homework

Listen to the audio-recording of your teaching episode. In your notebook, write down the questions that you asked.

In Principles to Actions, refer specifically to p. 37 (starting under Figure 14) and p. 39-40 Figure 16 “A comparison of questioning patterns…”

Examine the questions you used while playing the role of teacher and write responses to these prompts in your notebooks:

• Were your questions funneling or focusing questions?
• How might you change your questions now?
• What types of questions might be more productive for eliciting student thinking? Write a few new questions based on the 4 questions types on page 36 and 37.

Also: Tomorrow (Wednesday) we will be getting into your textbooks. Please bring materials to look at!

Percolation assignment: When is it “ok” to use funneling questions?

We started the morning by making posters for the 8 “rules that expire” and having a short discussion about those rules. We handed out the article “13 Rules that Expire” from Teaching Children Mathematics. (Don’t forget that the “Nix the Tricks” download is available on our Google folder. You can also visit the website and submit your own ideas for “tricks” or “vocabulary” that can be problematic when teaching children.) Then teachers discussed their addition and subtraction with signed numbers word problems that they wrote for HW last night.

We began the math discussion by wrapping up a loose end from yesterday.  We used skip-counting and pattern recognition to justify the rules of sign for multiplication.

We then moved into Unit 5 on Division.  The first major idea is the dependence of division upon multiplication, like how subtraction is depended upon addition.  This has two ramifications: (1) a student’s mastery of multiplication has direct bearing on their potential to master division, and (2) we can justify properties of division by relating it to a corresponding multiplication problem.  We explored item (2) in depth as we saw why division by zero is problematic.  We skipped over the “Rules of Sign for Division” (page 5.10), but they are based directly on the rules of sign for multiplication.  We also had a side trip where we discussed how subtraction and division fail to be associative.

Just before lunch, Andrea shared a worksheet presenting several alternative algorithms for multiplication. We explored lattice multiplication and discussed how place value and the distributive property play out in that representation. You are encouraged to explore any of the other algorithms shared on that page. Bonus points if you can explain the Russian Peasant method!

We also spent some time exploring rate calculations and the adjective-noun theme for division.  The website that expands the adjective-noun theme throughout the K-8 curriculum is: http://www.adjectivenounmath.com/.  We will look at rates and unit conversion is more depth in Functions and Algebra.

After lunch, we had small group and then large group discussion of the pedagogy HW from last night. After a small-group discussion, we compiled teachers’ impressions of the beliefs about teaching and learning that they inferred for Ms. R and Ms. F from the reading. The teachers then talked in their groups about “things teachers can do to promote productive struggle in math class” and we compiled those as a whole group (we will continue to add to this list of strategies and then post the final list in the google folder at the end of next week).

Fran then shared her work from the Library Fines problem from yesterday – modeling productive struggle (she went down a number of dead end pathways, used a number of different representations – pictures, numbers, expressions).

Then we watched a video that showed a teacher using “My Favorite No” activity in her class as a way to promote productive struggle and risk-taking: https://www.teachingchannel.org/videos/class-warm-up-routine

We spent the remainder of the afternoon exploring partitive and quotitive division problems.   The core difference is what information is given.  The total number of objects is the dividend (first number) in both cases, but in partitive division the divisor is the number of parts the total is split into, while in quotitive division the divisor is the size of each part.  We saw how quotitive division can also be called “measurement division” since it appears in conjunction with a nonstandard measure (e.g., “A pencil is 8 inches and a paperclip is 2 inches.  How many paper-clips-long is the pencil?”).  Beth provided another name for partitive division,”dividing to split,” and quotitive division, “dividing to fit.”

At the very end, Fran and Andrea discussed the Common Core Standards for Mathematical Practice.

Math Homework

1. Finish the problems from “Exercises of Partitive and Quotitive Division of Whole Numbers”
2. Problem 6 from “Practice Problems for Division,” page 5.16.  Actually making the paper rectangle is optional.
3. The handshake problem.  Problem 5, page 2.11.

Pedagogy Homework

As a community, we have focused on establishing a number of norms (ways of working) this week.  We have also focused on a few messages about the learning and teaching of mathematics. In your notebook, please write a response to the following prompts from your perspective as a PMI workshop participant. There are no right or wrong answers to these prompts – we want you to respond from your perspective.

1. Briefly describe three (3) different norms that we have established about participating in the workshop activities.
2. Briefly describe two (2) different norms we have established about being a productive group member.
3. Briefly describe two (2) messages that you have taken from the workshop (thus far) about what it means to “know” and “do” math.
4. Briefly describe two (2) messages that you have taken from the workshop (thus far) about teaching math.

We began the morning with teachers filling out the HW chart (Choosing Problems for HW Review on the HW Review Strategies list we’ll hand out at the end of the workshop). Teachers found a partner that they had not worked with already this week and compared/contrasted HW solutions.

After the homework review we moved into a discussion of signed numbers.  We explored how our models of addition and subtraction, particularly the chip model and number line models, extend to signed numbers.  We found that viewing subtraction as missing-addend addition (e.g., translating “a-b=[what?]” into “b + [what?] = a”) is very productive viewpoint when signed numbers are involved.  As a side diversion we discussed pitfalls of the word “cancellation,” and how that gives students trouble later in algebra.

Just before lunch, we watched a video of Carol Dweck (author of Mindsets) found at this URL:  https://www.youtube.com/watch?v=NWv1VdDeoRY

After lunch, we talked about conceptual understanding and procedural fluency (last night’s reading assignment). Fran led the group in making an iceberg model for 3X4=12.

Then teachers worked in grade-band groups to create iceberg models for: comparing fractions, place value, addition to 20, and the coordinate plane. At the end of this session we watched another short video about praise and mindsets: https://www.youtube.com/watch?v=hiiEeMN7vbQ

We then moved into Unit 4 and discussed multiplication and the distributive property.  We started with three models of multiplication: repeated addition, skip counting, and area.  We saw how repeated addition can give rise to each of the other models.  We also discussed how each model has its own limitations and strengths.  We then used those models to help us understand why the commutative and associative properties are true for multiplication, as well as why 1 and 0 have the multiplicative properties that they do.

We also discussed how the distributive property describes the interaction of multiplication and addition, and applied it to calculations.  We briefly connected “pulling out common factors” to reading the distributive property in reverse.  We also looked at how the distributive rule works more generally than what FOIL (“First Inner Outer Last”) can describe.

Math Homework

Read pages 3.28-3.32 on the opposite of a number and how it can be useful.  This likely matches with whatever you remember of how to add and subtract signed numbers.

Write word problems whose calculations involve the sums on page 3.24.  Try to do the same for the subtraction calculations on page 3.27.

Pedagogy Homework

Read Principles to Action, the section titled “Support Productive Struggle in Learning Mathematics” (p. 48-52).

In your notebook, write a response to this prompt:

• Review the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles). What beliefs are evident in Ms. Flahive’s and Ms. Ramirez’s classrooms (see fig. 21 on page 51)? What impact do those beliefs have on students’ opportunities to grapple with the mathematical ideas and relationships in the problem?