“Math as a Second Language” continues!

We began the morning by looking at the homework chart participants completed as they came into the room. For each of the homework problems, participants chose one of these four responses to mark: 1) I had no difficulty solving this problem; 2) I was stuck but working with someone helped; 3) I got stuck but figured it out by myself; or 4) I got stuck and working with others did not help. Based on the responses, participants spent a few minutes working on #12 or #13 in table groups. Participants then worked a spent a few minutes on reviewing answers to the Gauss Library problem. We moved from the discussions about homework to a review of homework from Monday night about Levels of Cognitive Demand (LoCD). In their table groups, participants spent 3 minutes each describing the LoCD of the tasks in a typical lesson and on a typical homework assignment. Gwen and Fran then talked to the whole group about the importance of providing students with the opportunity to engage in all levels of thinking represented by this framework and how teachers use this framework in making instructional decisions. For example, if a text has a majority of tasks that require a low LoCD, then a teacher needs to be conscious of providing other opportunities for students to engage in the kinds of thinking required by high-level tasks. Following this discussion, participants spent the last 30 minutes creating “iceberg” representations (ala the article they read for homework) for a particular K-6 skill or procedure.

We then continued into our discussion of why “the rules” of addition and subtraction of signed numbers work make sense in terms of the models we previously discussed. Along the way we bettered our understanding of how the models for addition and subtract work even for the natural numbers.

Following lunch we moved into Unit 4 (Perspectives of Multiplication) by discussing the parallels between addition and multiplication. We found the same four rules have versions for both addition and multiplication: commutativity, associativity, identity (0 for addition, 1 for multiplication), and inverse (which we call “opposite” for addition and “reciprocal” for multiplication). We also justified commutativity and associativity in terms of the repeated addition model and area model for multiplication before moving on to a comparison between the relative strengths and weaknesses of those two models (repeated addition may be more intuitive, but cannot handle non-whole numbers as easily).

After a break we moved into a discussion of the distributive property. This forms a ninth rule for arithmetic, to combine with the 4 rules (each) for addition and multiplication. We showed how this rule makes intuitive sense in terms of both the repeated addition model and the area model.  The area model also shed light on expressions like (a+b)*(c+d+e) and expanded them into ac+ad+ae+bc+bd+be.  We also used the distributive law to make sense of lattice multiplication and a few multiplication “tricks.”

Handouts

• Solution to “Carl Friedrich Gauss Library Overdue Fees”
• Solutions for Unit 2
• Unit 4: Perspectives on Multiplication
  • We will skip the content of page 4.12

Homework

• [No reading homework]
• Complete the survey Fran distributed.  (Download a new copy here.)
• Revisit problems 4 and 7 from 3.40 in light of our discussion of signed numbers.
• Try to find some sense in the multiplication rules for negative numbers by completing pages 4.20-4.22.
• Enjoy State College on a beautiful evening.