# Blog Recap: 2019 Day 9

We began the day with some more work on linear and proportional relationships through the Purple Paint problems. Then we did Burning the Candle problems. We were especially interested in graphical representations of these scenarios and connected graphical representations to one another (red to blue; blue to red; total to red) and to other representations (tabular, symbolic, contextual). Then we discussed inverse relationships and we concluded by discussing at a “meta” level what things we might be interested in when describing “co-variation.”

After lunch we followed a jigsaw arrangement to discuss the 4 messages (readings are attached). These all come from Cathy Seeley’s messages books. Go here for more info about these books.

Then we spent time looking in our curricular materials, looking for ways we might start the year off well by raising the tasks’ level of cognitive demand, letting things go, incorporating review INTO new materials, supporting productive struggle, etc.

Then we classified sequence problems as linear, proportional, or neither, in order to better appreciate their attributes.

FIS ch. 16

FIS ch. 14

Let it Go..

Effectiveness and Efficiency

# Blog Recap: 2019 Day 5

Today we

This morning we talked about multiplying fractions using the gardening plot problem. We discussed division, including division by zero.

Then we talked about talking about some “Rules that Expire,” but we didn’t actually talk about the rules themselves. đ

After lunch, we explored the NCTM website and resources. Then we discussed “Why can’t kids remember their basic facts?” and we read and discussed ways to help kids learn these facts. We talked about the developmental nature of learning facts (3 phases), the importance of using problem solving and strategies we can teach kids to help them reason.

We concluded with the modes of addition and subtraction and jogging and cutting cloth tasks for fraction division. (And how to “avoid” division).

We said good-bye to many new friends and colleagues, and we heard some words of wisdom from each of them.

# Blog Recap: 2019 Day 3

This morning we:

1. Discussed the “sum of odds” problem (1 + 3 + 5 + …….+ 99)
2. Built on that to work on the “Insatiable Caterpillar” problem
3. Then we worked on the Piggy Bank Problems
4. Then we played (and nailed!) the name game

After lunch we:

Discussed how developing conceptual understanding can support procedural fluency. We generated the list below:

• âConcrete to abstractâ : concrete action supports conceptual understanding, and this provides access to be able to model themselves
• Hands-on activities –> mental representations –> supports abilities to do procedures
• If only procedural skills: kids either remember or they donât, and then donât have resources
• âtransfersâ to other problems, other representations. THIS IS MATHEMATICAL THINKING
• Our job is to create problem-solvers. We have so many tools, kids need to know how and when and why to use them.
• Provides meaning (beyond âjust for the testâ)

Watched video of Rachel’s renaming of a mixed number into an “improper fraction”

Then we developed iceberg posters representing concepts that build “floating capacity” for a variety of procedures.

We ended the day by naming the properties of multiplication. This related to today’s mathematical themes: There are many multiplication algorithms, each with their own advantages, and The methods of decomposing and recomposing are especially key to making sense of multiplication algorithms.

Tonight’s homework:

Read Principles to Actions section on productive struggle (pp 48-52).

Read the article we distributed called Supporting productive struggle with communication moves.

Spend 20 minutes on the “Handshake problem” on the Evening Reflection page of Day 3.

# Blog Recap: 2019 Day 2

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 PMIâs 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 oneâs 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:

1. Putting into real-life situation
2. Use âjustifyâ and âexplainâ in problem
3. Open up multiple ways to do the problem
6. Use manipulatives or hands-on
7. Use compare and contrast
9. Use questions where there is not just one right answers (allow multiple correct

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)

# Daily Recap: Week 1 Friday

We began by continuing our discussion of division with fractions. We drew pictures and looked for patterns in an effort to understand why it works to “invert and multiply.”

Then we reflected on the reading about “Upside-Down Teaching,” and brainstormed ways to shift from I-We-You approaches to You-We-I approaches.

We had time to plan for this shift by looking at curricular materials and plans for the first few weeks of school. We watched a video of the “My favorite no” practice.

We ate a delicious lunch together and some of us said good-bye to Fran. Sadface.

After lunch, Melina shared recommendations for the OGAP frameworks. Thanks, Melina!

We resumed work on operations with fractions. This time we pushed ourselves to develop even deeper understandings of the “invert and multiply” procedure.

# Daily Recap: Week 1 Thursday

We began by revisiting the “brownie” homework from Tuesday night, which called for us to critique student responses in light of “the whole.”Â  This made us sensitive to the issue accounting for “the whole” as we worked through the Garden Plot and Land-Tax Task to illuminate “the rules” for multiplying fractional quantities.Â  In the end we came to the following conclusions (among others):

• TheÂ array andÂ area models introduced in grade 3 directly impact how one can understand fraction multiplication in grade 4.
• The numerator functions to count the “number of bits” and the denominator indicates size of the “bits” in terms of “how many bits to make 1 unit” (which may or may not be the total number of bits shown).Â  Unlike the brownie problem, “the whole” is not arbitrarily chosen but rather is a consequence of “the unit” for each of the factors.

After lunch, we first created a list with this sentence stem: To support productive struggle, teachers can…

Then we discussed “learning goals” and “performance goals” (see handout from today). After that, we worked in grade-alike pairs/groups to plan for the implementation of a high-level task and supporting productive struggle.

Then we watched a video about mindsets and the impact of different kinds of praise. And Fran shared Jo Boaler’s book on mathematical mindsets:

Melina suggested that we check Amazon for grade-level books by Jo Boaler. Here is the link to the 3rd-grade book.

Following the discussion on productive struggle, we struggled productive ourselves as we grappled with the “Cut from the same cloth” problem, division, and how natural division of fractions is.Â  At the tail end I mentioned the other post with good sources of rich problems.

Pedagogy HW: Read “Upside Down Teaching” by Cathy Seeley (handout).

• Think about how this message connects to the teaching practices that we have talked about this week from Principles to Actions: Implement tasks that promote reasoning and problem solving; Use and connect mathematical representations; Build procedural fluency from conceptual understanding; and Support productive struggle in learning mathematics.Â  Make some notes to support your participation in group discussion tomorrow.

Math HW: Reflect on Division of Fractions and why the “keep-change-flip” trick is accurate (but short-cuts the conceptual underpinning).

# Blog Recap: Week 1 Monday (7/16/18)

A great first day!

We started with introductions, along with welcome messages from Mark Levi and George Andrews.

We then jumped into our first problem, theÂ Star Spangled Banner Problem.Â  Participants shared a variety of solution methods.Â  These led to the following formulas:

Formulas for the number of stars after n days

We briefly touched on connections between these, and will return to that later.

We spent the rest of the morning on the pre-test and then participants got a chance to share their backgrounds and start networking before lunch.

After lunch, we discussed “Smarter than We Think” (by Cathy Seeley). We developed posters that listed “inspirations” and “challenges.”

Fran presented three frameworks that guide PMI’s 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 oneâs own efficacy.

• 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 ended the day with a discussion of the “Adjective Noun Theme,” which is based on the notion that numbers are most useful when they are considered with context.Â  Numbers areÂ adjectives describe the amount of some noun.Â  We then tried to apply this viewpoint to compare similar fractions, treating the denominator as a noun and the numerator as an adjective.

Math HW:

Refer to “Rules that Expire” (Block 1, p. 15 of notebook). For your assigned rules, follow the instructions on the sheet.

Pedagogy HW:

Read Principles to Actions (the blue book): Sections titled âProgress and Changeâ and âEffective Teaching & Learningâ(Pages 1-12)

• What did you learn from reading the âProgress and Changeâ section about the state of mathematics education in the U.S.? Write a couple of thoughts in your notebook.
• Think about: How did you react to the chart of beliefs on page 11? In your notebook, write a reflective response (a few sentences) to the beliefs chart.

# PMI@UP 2017 Day 10

What a full two weeks! Â And yet there always seems to be more to do.

We started the morning with the post-assessment and beliefs survey. Â The analysis of that data will be shared later this summer.

We then spent some time refreshing the addition and subtraction of signed numbers, and representing that with joining and taking away piles of color-coded counters. Â That discussion progressed into exploring theÂ multiplication of signed numbers. Â We found that when one or two of the factors are positive whole numbers, then we can rely on the “__ groups of size ___” model. Â However, when both factors are negative we need to rely on logic and the distributive property in particular. Â We ultimately had three explanations: one based on patterns, one based on the notion of the opposite, and one based on comparing the products 4×4 and the expanded form of (5+(-1))x(5+(-1)).

We then split into teams of 3 to do “Bungee Barbie” and started analyzing the data over lunch. Â Here is the winning drop: Â https://photos.app.goo.gl/TJO0BwptyT0slWrm2Â  . Â After the drops, we debriefed on the mathematical thinking and topics that occurred within that task.

We then got into our “Kumbaya” circle and reflected on what we learned about mathematics, what we learned about teaching, and our commitments to how we can improve our own classrooms. Â We then read “Hooray for Diffendoofer Day.” Â The key take-away message: Teach your students to think and the tests will take care of themselves.

### Homework

ContinueÂ to give your students the best educational experiences possible. Â Continue to learn how to make those experiences richer, and continue to seek outÂ why of mathematics.

# PMI@UP Summer 2017 Day 1

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).

1. Conceptual Understanding â comprehension of mathematical concepts, operations, and relations.
2. Procedural Fluency â skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
3. Adaptive Reasoning â capacity for logical thought, reflection, explanation, and justification.
4. Strategic Competence â Â ability to formulate, represent, and solve mathematical problems.
5. Productive Disposition â habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and oneâs own efficacy.

Homework

Read Principles to Actions (the blue book):

1. Sections titled Progress and Change; Effective Teaching & Learning (Pages 1-12)
2. Section titled âImplement Tasks that Promote Reasoning and Problem Solvingâ (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), 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?
• Consider the chart about beliefs on page 11. What is your reaction to this description of unproductive and productive beliefs?

Read through and reflect on the 8 Rules that Expire.

• When does the rule work?
• Bonus: What grade level does that scenario occur?
• Give a mathematical scenario where the rule does not work (at least, not in the way it’s phrased)
• Bonus: What grade level does that scenario occur?

# PMI@Berks Day 4 (6/29/17)

It’s hard to believe our week is almost over.

For Friday:
Read:Â Principles to Actions sections on “Facilitate Meaningful Mathematical Discourse” and “Pose Purposeful Questions,” pages 29-41.
Do:Â “Problems on Division” Block 3, Page 37. Â  Problems 3 and 4.