Author Archives: Andrea McCloskey

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 8

Today we talked about unit rates (and representations thereof) based work with the Coffee Maker problem and the Perfect Pint of Pink Paint. See the image below.

Then after lunch, we discussed the reading about “access and equity” from Principles to Actions. We discussed productive vs unproductive beliefs, we played with the scenario cards, and we connected all of that with

  • the 5 strands of mathematical proficiency
  • the Standards for Mathematical Practice
  • The Effective Teaching Practices

Then we solved the Hourglass problem, and compared student work thereon.

Tonight you are invited to pre-read your assigned “message” reading. These were distributed at the end of class.

Have a good night and see you tomorrow!


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. 🙁

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:

  • ”Concrete to abstract” : 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 don’t, and then don’t have resources
  • “transfers” 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 ”just for the test”)

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 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 – comprehension of mathematical concepts, operations, and relations.
    • Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
    • Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification.
    • Strategic Competence –  ability to formulate, represent, and solve mathematical problems.
    • Productive Disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.This book can be downloaded for free at:
  • 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 “justify” and “explain” in problem
  3. Open up multiple ways to do the problem
  4. Giving the answer and asking how you could “find” this answer (more than one
  5. Asking “how” questions instead of just “what” questions
  6. Use manipulatives or hands-on
  7. Use compare and contrast
  8. Make it less about “answer getting” and more about “math thinking”
  9. Use questions where there is not just one right answers (allow multiple correct
  10. Asking students to find “one more” 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.  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.  MTMS, 2008.  (Handout from today)

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


Daily Recap: Week 2 Thursday (7/26/18)

Today we began by discussing the Coloring Fun! (yellow-red-blue patterns) “Burning the candle” problems.

After lunch we made a giant iceberg poster representing how ideas at each of our grade levels build floating capacity for understanding linear relationships such as those in the candle burning problem.

Then we spent some time exploring and sharing resources we found on the NCTM website with our new memberships!

Then we talked more about other kinds of relationships between quantities besides proportional: linear and non-linear alike!

For homework:

Read the Professionalism Section of Principles to Actions (pp. 99- 108).


Daily Recap: Week 2 Monday

This morning we began by discussing algebraic reasoning and reasoning with patterns. We used the Sibling Rivalry problem to illustrate how we can reason around an unknown baseline.

We then discussed modeling positive and negative numbers using a number line, addition and subtraction on that number line, and then we used colored chips to do the same. Thanks, Derrick!

After lunch, we discussed timed tests and teaching basic facts. We read together from this selection from Van de Walle, Karp, and Bay-Williams. Andrea recommended two articles from Teaching Children Mathematics: one about assessing basic facts by Kling and Bay-Williams, and  one about the importance of thinking by Buchholz.


From Principles to Actions (the blue book): Read two sections:

  1. Pose purposeful questions (p. 35)
  2. Elicit and use evidence of student thinking (p. 53)

In your notebook, write a response to these three prompts:

  1. In questioning small groups of students working on a problem, a teacher noticed that when she asked a “focusing” question, the students continued to look at their work and continued to engage in their own dialogue. When she asked a “funneling” question, the students looked up at the teacher. Comment on these observations.
  2. Listen to your audiorecording from today. Use fig. 14 on p. 36 and fig. 16 on p. 39 to write a description of your question patterns.
  3. How might you change your questioning to elicit and then use evidence of your students’ thinking to move the student forward to the mathematical goal of the problem?


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.


PMI@UP Day 9 2017

Today we

Productively struggled our way through the candle burning problems as we learned about linear relationships

Watched some videos:

  1. Mindset #1:
  2. Mindset #2:
  3. Escalator:
  4. My favorite NO:

Worked in grade level groups to made a big iceberg poster about linear relationships. How do the concepts we teach at each grade level build floating capacity for engaging with the candle burning problem?

Discussed parents and families, using the message from Cathy Seeley’s book.

Discussed negative and positive numbers.

For tomorrow, please:

Read the Professionalism section of Principles to Actions (pp. 99- 108)

PMI@UP Summer 2017 Day 6

Today we

  1. Moved onto the algebraic thinking units
  2. Discussed basic facts, drills, and timed tests (see the messages we generated, below). Here is the website Andrea shared that provides opportunities for selected practice opportunities for fluency development
  3. Discussed calculators in the math classroom
  4. Interviewed a classmate for tonight’s homework

Here are some websites with calculator lessons and activities:

  1. The Math Tools website ( allows you to search by grade level and by the type of technology you wish to use.
  2. Texas Instruments has a collection of calculator activities to review ( calculators/downloads/US/Activities/).
  3. Casio has several calculator activities to review for elementary and middle school (

Here are our messages to ourselves about basic facts:

  • More work on strategies and practicing the strategies BEFORE drilling
  • When a 9th grader is struggling, provide tools for remediation (teach strategies!). It’s not enough to say “learn them!”
  • Timed tests are required by my district—use it as a learning tool (and assessment). Assess for individual fact families and find out what strategies they do/don’t use
  • Some facts are more powerful than others
  • Timed tests frustrate the struggling students even more and reward the quick thinkers
  • “you’re competing against yourself, not each other”—emphasize personal growth (chart for themselves)

BTW: Here is a nice website about number strings.

For tomorrow, please

Read Principles to Actions. “Pose Purposeful Questions” (p. 35-41) and “Elicit and Use Evidence of Student Thinking” (p. 53-57)

In your notebook, write a response to these three prompts:

  1. In questioning small groups of students working on a problem, a teacher noticed that when she asked a “focusing” question, the students continued to look at their work and continued to engage in their own dialogue. When she asked a “funneling” question, the students looked up at the teacher. Comment on these observations.
  2. Listen to your audiorecording from today. Use fig. 14 on p. 36 and fig. 16 on p. 39 to write a description of your question patterns.
  3. How might you change your questioning to elicit and then use evidence of your students’ thinking to move the student forward to the mathematical goal of the problem?

Try the Maze Playing Board. Let’s see who has the largest value tomorrow. There MAY be a prize involved.