Our math for the morning explored the theme that “the same quantities have many names, and context determines the most useful name.” We focused on discussions of the equal sign, place-value, and fractions and algorithms for computing operations of addition and subtraction.

In the afternoon, Fran presented three frameworks that guide PMIâs design and implementation:

**Strands of Mathematical Proficiency: What does it mean to ***know* mathematics? Here are the definitions of the Strands of Mathematical Proficiency that were in the PowerPoint today:*Adding it Up: Helping Children Learn Mathematics*, National Research Council (2001, p. 116).
- Conceptual Understanding â comprehension of mathematical concepts, operations, and relations.
- Procedural Fluency â skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
- Adaptive Reasoning â capacity for logical thought, reflection, explanation, and justification.
- Strategic Competence â Â ability to formulate, represent, and solve mathematical problems.
- Productive Disposition â habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and oneâs own efficacy.This book can be downloaded for free at:Â http://www.nap.edu/catalog/9822/adding-it-up-helping-children-learn-mathematics

**Standards for Mathematical Practices: What kinds of activities should students engage in while learning mathematics? **These are listed in *Principles to Actions* and defined on in your notebook in the front matter section on page 5.
**Eight Teaching Practices: How should teachers teach so that students engage in learning mathematics so that they become mathematical proficient? **This is the main content of *Principles to Actions: Ensuring Mathematical Success for All. *We will unpack several of these teaching practices over the two weeks!

We also read the *Principles to Actions* section about “implementing tasks that promote reasoning and problem solving.” We discussed the levels of cognitive demand, and we generated a group list of strategies for tweaking low level tasks to raise their level of cognitive demand:

- Putting into real-life situation
- Use âjustifyâ and âexplainâ in problem
- Open up multiple ways to do the problem
- Giving the answer and asking how you could âfindâ this answer (more than one

answer)
- Asking âhowâ questions instead of just âwhatâ questions
- Use manipulatives or hands-on
- Use compare and contrast
- Make it less about âanswer gettingâ and more about âmath thinkingâ
- Use questions where there is not just one right answers (allow multiple correct

answers)
- Asking students to find âone moreâ way or answer

During our math for the afternoon, we explored combining collections of even and odd numbers.

**Math HW:**

On the Day 2 Evening Reflections, do everything but items #1 and #3.Â You also don’t need to do the “aliases of a fraction” prompt. [note: by “Reference” pages, Andrew means pages 4 and 5 from the packet.]

**Pedagogy HW**:

Read Principles to Actions: Section titled *Build Procedural Fluency from Conceptual Understanding* (pp. 42-48).

Also read: Webb, Bozwinkel, & Decker. *Beneath the Tip of the Iceberg: **Using Representations to Support Student Understanding.*Â MTMS, 2008.Â (Handout from today)

Think about how these two readings connect and make some notes in your notebook.