Monthly Archives: July 2014

Blog Recap: Day 4 of Mathematics as a Second Language

Get better acquainted with others in the group.  Showing the NCTM and what it offers, such as the publication “Teaching Children Mathematics.”  We did a “Gallery-walk” for the iceberg posters, adding sticky notes with comments.

After the morning break we dug into the rules of sign for multiplication of integers in several ways.  Hopefully we shed light on a few ways of why two negatives should multiply to make a positive.

We then spent some time exploring some of the peculiarities of division, and considered how the many faces of division lead to confusion.  After lunch we explored division in light of its being “un-multiplication” and how fact families with multiplication can make sense of rules like “you can’t divide by zero” and how the signs work for integer division.

At the end of the day we also consider how the repeated addition and area models for multiplication can provide models of division, together with the number line model of splitting a length into smaller (equally-long) segments.  We also classified division problems according to whether they were “partitive” (i.e., we are given the number of groups and want to know how big the groups will be) and “quotitive” (i.e., we were given the size of the groups and want to know how groups can be made).  [Not discussed in class, but relevant: the terms “partitive” and “quotitive” can be seen to refer to the divisor.  In partitive division the divisor describes the number of parts, while in quotitive division the divisor describes “how many per group” (like a quota).


  • Unit 5: The Many Faces of Division (28 pages)


  • “Practice Problems for Division” (pp. 5.17-19)
    • Problem 3.  Try to explain your method of solution in at least two different ways.  (Note: “way” is not the same as “showing different work.”  Two different mindsets can still produce the same work.)
    • Problem 5ab: Evaluating the division to a numerical value or a simplified form itself is not the point.  It’s the process of conversion.
    • Problem 10.
  • Solve: “Fran has a 5-foot ribbon and wants to cut it into 10-inch segments.  How many segments will she get?”

Blog Recap: Day 3 of Mathematics as a Second Language

“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.”


  • 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


  • [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.


Blog Recap: Day 2 of Mathematics as a Second Language

Welcome back to Day 2!

Discussed Film Developing Problem, both the methods of solution and classifying it according to lower or higher levels of cognitive demand.  Moved into groups based on grade-bands (K-2, 3-4, 5-6).

Commutativity and Associativity of Addition, and how to illustrate it with models of addition.   The sum 1+2+3+…+100 (triangular numbers).


Start of Unit 3: The Art of Unadding.  Approaches to subtraction; the adjective-noun theme (Fundamental Principle of Subtraction); alternate algorithms for subtraction.

Features of the number line (including negative numbers). Applying models to the arithmetic of signed numbers.

By the way, if you’re curious about how old certain mathematical symbols or terms are, see: and


  • Daily Schedule for Week 1
  • Solutions for Unit 1
  • Unit 3: The Art of Unadding (42 pages)
      • We discussed the material (although not along with the sheet) on pages 3.1-3.14 and 3.26-3.37.  We skipped the material on “Processes” on pages 3.17-3.25.


  • Reading: Principles to Action, Section “Build Procedural Fluency from Conceptual Understanding” (pp. 42–48).
  • Reading: Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding.  MTMS, 2008.  (Handout from class today).
  • “The Carl Friedrich Gauss Library Overdue Fees” (The Variation can be viewed as a challenge.)
  • “Additional Exercises on Subtraction,” page 3.40.  Problems 4, 7, and 12 or 13.

Blog Recap: Day 1 of Math as a Second Language.

Welcome to PMI, everyone!  We will be doing these blog recaps at the end of every day,  both to list the handouts you received, the homework for the next day, and to keep track of how far we’ve come.  Please add your comments to this post and keep the discussion going!

We started with the welcomes from Math Department Chair Yuxi Zheng, along with the director of PMI George Andrews.   Then we moved into the first problem-session: “Kayaking on Lake Champlain,” and saw how the same problem can be solved in a variety of ways (e.g., charts, graphs, and algebra), and even how the same problem can be interpreted in multiple ways (e.g., round up or not?).  Next was a discussion of productive versus unproductive attitudes about mathematics, which lead into a debate about what it means to be fluent versus being efficient.  Be sure to keep an eye out for your own habits of mind and what attitudes you hold.

Following a break we had the Pre-Assessment, and then lunch.

In the afternoon we started up with a discussion of equality and the equals sign, along with chains of equalities and what it means for two algebraic expressions to be equal.  After the break we moved into Unit 2, introducing the adjective-noun theme and the Fundamental Principle of Addition (add the adjectives and keep the nouns).  We then explored the consequences of the Fundamental Principle of Addition as it applies to place-value and fractions.

Handouts from today:

  • Front Matter: Cover page, Course Overview, Course Expectations
  • Unit 1: Fundamentals (21 pages)
    • We will be skipping pages 1.4, 1.5, 1.10, 1.15, 1.16, 1.18, 1.19, 1.20, 1.21.
  • Unit 2: Perspectives on Addition (14 pages)
    • We will be skipping page 2.14.  As of today we’ve discussed pages 2.1–2.7.


  • Read the Course Overview and Learning Objectives.
  • The Film Developing Problem (Page 1.2)
    • Find multiple solutions to the Film Developing Problem
    • How is this problem similar to the Kayaking problem?  How is it different?
  • Read Principles to Action: Section titled “Implement Tasks that Promote Reasoning and Problem Solving” (pages 17-24).
    • Reflect on a typical in-class math lesson (use your textbook to refresh your memory). Using the descriptions of Levels of Cognitive Demand in Figure 3 (p. 18), be able to describe the types of mathematical thinking your students are required to engage in during a typical lesson.
    • Reflect on a typical homework assignment that your students complete (use your textbook to refresh your memory). What level of cognitive demand do most of the tasks on a typical homework require of your students?
  • Guess my noun (Page 2.7), first 5 problems (or more if you like)