Recap: Mathematics as a Second Language Day 4 (July 23, 2015)

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:  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:

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.


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