Day 4 of Math as a Second Language (7/21/2016)

Rules that Expire

We began the morning with “Rules that Expire” – we had 10 rules and groups of teachers made a presentation about their particular rule that expires. All of the rules came from an article published in Teaching Children Mathematics titled “13 Rules that Expire,” which we gave to each teacher.

Rules of Sign

We revisited the why of (Negative)x(Negative)=(Positive).  Andrew shared two arguments (one from Matt), Fran shared a third, and Hope opened the discussion to introduce a fourth.  These four arguments were all based on the principle that arithmetic needs to be consistent, rather than cute analogies or contrived scenarios.

Unit 5: Division as Unmultiplication

We next moved into discussing division, starting with “walking back” the concept of how division is treated in fifth grade and then working grade-by-grade back to Kindergarten.  We found how the seeds of division are planted as early as Kindergarten.   We based our understanding of division on “unmultiplication,” specifically how any division calculation can be seen as filling in a missing factor for a multiplication sentence.  This perspective allows us to leverage our models of multiplication to inform how we view division.


Pedagogy: Productive Struggle Discussion

We began by compiling a list about beliefs about teaching/learning mathematics we interpreted from the description of Ms. Flahive’s and Ms. Rameriz’s classrooms (in the P to A reading). We then compiled a list of strategies a teacher can employ to support productive struggle, including:

  • create an atmosphere where it’s ok to make mistakes (and can help get the parents on board with this idea, too)
  • create and enforce effective boundaries and constraints with time
  • model their own thinking processes and their own math struggles
  • provide a mix of high-demand and low-demand tasks
  • provide a variety of tools and materials so that students can solve problems in different ways
  • allow students to communicate with each other and with you
  • require an explanation and not just an answer (students should expect to do this!)
  • prioritize hard work over natural ability/intelligence (growth vs. fixed mindset)
  • provide suggestions—not answers—to help students keep working productively (facilitating learning)
  • provide resources (like a poster) showing suggestions for getting “unstuck”
  • validate that you understand this is challenging
  • keep an open mind—be open to learning and trying new things
  • provide immediate feedback based on effort—praise effort (even if it’s only 1 part of a problem—find what they did do correctly)

We then split up by grade level and found (or created) a task that requires a high level of cognitive demand and began to plan for implementation, with a specific focus on supporting productive struggle.

We ended pedagogy time today by watching a video about mindsets:

Rates, and the Adjective-Noun theme for Multiplication and Division

We discussed how multiplication can be seen in conjunction with rates, such as how paying 2 dollars/hour for 3 hours of parking costs $6 since 2×3=6.  Going the opposite direction, paying $6 for 3 hours of parking means that the rate was 2 dollars/hour since 6÷3=2.

Rates play a fundamental role in grades 6, 7, and beyond.

Fraction Fundamentals

We next moved into a discussion of how participants teach fractions and how they themselves understand them.  We found myriad ways to represent a fraction as simple as 2/3 (see Andrew’s summary image below), but also found representations like “2 out of 3” to be limited when moving to improper fractions like 5/3.  Other models, and especially the number line model, work just as well for proper fractions as for improper fractions.


We closed with a discussion of equivalent fractions and the idea of “dividing to one” to justify when two fractions have equal value.  In short, we used area models to justify why 2/3 could be changed to 8/12 by cutting the rectangle to even smaller pieces and using more of them; then we read that equivalence backward to justify why 8/12 could be simplified to 2/3.  (We also discussed the pitfalls of using words like “reducing fractions” and “cancellation.)


Math Content:

Page 5.16, problem 6.
Page 6.18, problem 1&2.


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. Please respond 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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