# Recap: Mathematics as a Second Language, Day 3 (July 22, 2015)

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?