Author Archives: Fran Arbaugh

Blog Recap: 2019 Day 7

This morning we spent learning about why the integer (positive and negative numbers) operations work! We talked about adding, subtracting, and multiplying signed numbers – ultimately learning why a negative number times a negative number is a positive number!!!

We then embarked on solving the Kayak problem. After lunch, we continued with the Kayak problem before moving on to pedagogy.

During the pedagogy time today, we read Cathy Seeley’s message titled “Upside Down Teaching” and talked about positive reactions to the message and challenges with teaching in this way. We then did the “Planning for Implementing High-Level Tasks” activity.

For those who want to reread, tomorrow we will be reading and discussing the Access and Equity section in PtoA (p. 59-69).

Blog Recap: 2019 Day 6

In the morning, we welcomed new teachers and started thinking algebraically! We revisted Otto and Hannah (Sibling Rivalry: Unknown Quantities), but this time they were eating Halloween candy, comparing heights, and running!

After lunch, we learned about performance and learning goals (Band Concert Task) and revisited how to use goals to support asking assessing/advancing questions (like we did last week with the Brownie Problem). We then did an activity where we analyzed a set of student work (Walking from School Task) and discussed how goals can support selecting, sequencing, and connecting student work in mathematical discussions (see page 30 of PtoA and the entire section on Facilitate Meaningful Mathematical Discourse). Fran ended the session with this diagram, which shows how goals can inform all of the other 7 PtoA teaching practices:

We ended the day with the “Back and Forth along Euclid Avenue” task – exploring positive and negative numbers! We then did Block 1, page 4 “Adding and Subtracting Signed Numbers via the Number Line.”

Blog Recap: Week 1 Tuesday (7/17/18)

We began by sharing our thoughts on “Rules that Expire.”   (Posters: Rules that Expire)

We then moved to the issue of equivalent fractions, “cancelling,” and how to make sense of a fraction you must make sense of “the one.”

After lunch, we first met in small groups to discuss the Principles to Actions reading from last night and then debriefed as a whole group.

We then complete the following:

  1. Read Principles to Actions (the blue book): Section titled “Implement Tasks that Promote Reasoning and Problem Solving” (pages 17-24). Write answers to the following prompts in your journal:
  • 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?

Then we watched Dan Meyer’s TED talk.

Fran shared a resource for 3 Act Tasks, developed by Dan Meyer.

Next we talked about the variety of forms that addition and subtraction scenarios can take.  Next we started on the Insatiable Caterpillar task, which we will wrap up tomorrow.

Math HW: Analyze the student work on Block 1, Pages 13-14.  Think about the caterpillar.

Pedagogy HW:

  • Read Principles to Actions: Section titled “Build Procedural Fluency from Conceptual Understanding” (p. 42-48).
  • Also read: Webb, Bozwinkel, & Decker. Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding.  MTMS, 2008.  (Handout from today)

Respond to this prompt in your notebook: What connections do you see between these two readings? What questions do you have?

Here are three articles that you might find interesting to read and share with colleagues:


Day 5 of Functions & Algebra (7/29/16)

We started the morning with Andrea and Fran sharing resources. Andrea showed the Google Folder titled PMI 2016 Resources for Participants. Then she showed the NCTM website, and discussed benefits of membership (remember that K-8 institutions get a great deal!!). This segment ended with Andrea and Fran sharing some books (here are the covers):


Andrew and Matt spent a brief time discussing Unit FA-4 on Functions, which are really just a formalization of the Processes from FA-1 with a new notation.  We then dipped into the Fox’s Furniture Store sequence from FA-5, both from the perspective of solving problems and evaluating students work on its mathematical merits.

Next we formed a large circle where Fran led a discussion about the readings on Professionalism from Principles to Actions.

After lunch, the group took the post-test (a necessary evil).  After that wrapped up Andrew and Matt tied up some loose ends and lingering questions: the Chipmunk formula from Wednesday’s homework, a strategy for solving the Milk Problem, and a justification for the “add the digits” trick for recognizing multiples of 9

It has been a fantastic two weeks!  We will be writing a longer post later with a full summary of these workshops and the workshops at Greater Allegheny and Brandywine as well.


Keep up your enthusiasm.  Seek change, and be patient.

Keep the commitment you made to yourself as you progress through the school year.

Don’t shy away from digging for the why of the mathematics you teach.  It all hangs together to form a cohesive whole.  You are welcome to send burning mathematical questions to Andrew.

Look for good candidates to recommend to PMI next summer.

Keep us updated on any revelations or experiences in your classroom that you can attribute to PMI.

Day 3 of Functions & Algebra (7/27/16)

We are hip-deep in linear relationships (and a few things that aren’t).

Gallery Walk

We started the morning with posting some of the extension problems that participants wrote last night and then did a gallery walk.  Andrew highlighted two strategies for getting students to move to systematic or efficient strategies: using uglier numbers that are less prone to “I just knew it,” and multi-step problems where the task requiring the strategy is repeated multiple times and is part of a larger whole.

The Milk Problem

Then Andrea, Matt, and Andrew led a discussion of the milk problem, trying to make 2% milk from 1% milk and 4% milk.  A mathematical take-away lesson is “proportions don’t add, but amounts do.”  A meta-cognition lesson is “sometimes you just need to try something to see why it doesn’t work.”

Currency Conversion

Then Andrew continued the Currency Conversion problem, where now there is a fee involved.  We drew a lot of nice lessons out of it already:  parallel lines correspond to lines with the  same slope, the conversion rate corresponds to the slope of the line, the benefits of writing both fee-schemes as “convert to euros then subtract fee.”

Upside down teaching, real-world problems

We then moved into a pedagogy session, where we began with a discussion of the rich problems teachers developed during HW. Then each table generated a list for tomorrow’s iceberg activity (also from HW prompt). We also had a discussion of upside down teaching (from last night’s reading). Finally, we watched Dan Meyer’s TED talk.

Andrew shared the site which has mathematically rich games and puzzles that students will happily engage in.

Terminology for Linear Relationships

Matt led a discussion putting names (slope, y-intercept, x-intercept) to concepts identified in the Currency Exchange with Fee problem.

The Tortoise and the Hare

We looked at the Tortoise and the Hare problem, wherein a scenario is presented and participants had to ask for certain information (e.g., the speeds) and not just handed it from the start.  Table-groups were then given different follow-up questions and asked to present their answers to the rest of the group.  Along the way we saw how many natural questions can be answered via a graph.

We finished the day be repeating the teaching episodes from Monday, but now practicing purposeful questioning and eliciting student thinking.

Homework for tonight:


Write formulas for the Tortoise and the Hare problem that can answer this question: After x minutes, how many feet has the tortoise/hare/chipmunk run?  The following information was provided during class:

  • In the first minute the tortoise ran 1200 feet
  • In the first minute of running the hare ran 3000 feet.
  • The tortoise and hare maintain a constant speed.
  • The hare starts with a 10 minute nap, then runs.  The tortoise starts right away.
  • (Follow-up #7): A chipmunk starts from 12,000 feet behind the tortoise and runs at 2000 feet/minute.


  1. Listen to the audio-recording of your “revised” teaching episode.
  2. In your notebook, write down the questions that you asked.
  3. Examine the questions you used while playing the role of teacher and write about what you learned about trying to use purposeful questions and eliciting student thinking.
  4. Read one of the messages from Smarter Than We Think. If your last name begins with A-M, read Message #14. If your last name begins with N-Z, read Message #16.
  5. Reflect back over all of the pedagogy discussions we have had in PMI and start a list in your notebook titled “Things I can do to help my students persevere when solving rich math problems.” Add at least three things to your list.

BTW: The “Messages” we’ve been reading come from the excellent book pictured below.




Recap: Functions and Algebra Day 3 (July 29, 2015)

Today we began by making vocabulary posters for the terms proportion, x-intercept, y-intercept, linear relationship, and slope. Here are the directions for that activity (also included on the “PMI Strategies for HW Review” document in Google folder – look under “Gallery Walk”):

Using large posters, put one vocabulary term per sheet. Arrange the posters around the room. Give each student a set of small post-it notes. Have students visit each post-it, read what others have written, and add an idea, example, concept, etc. about that term. Once that process is complete, assign a small group of students to each poster and ask them to make categories out of the small sticky notes and be ready to talk about their categories as well as any sticky notes that they have a question about mathematical validity. Discuss the terms as a whole group.

We then reviewed the homework by putting the answers on the document camera.

We resumed our discussion of the Currency Conversion with Exchange Fee from yesterday afternoon.  Those solutions are now available in the Google Drive folder.  Problem 12 provided an opportunity to discuss algebra’s strategic advantage as a tool: equation-solving skills provide an efficient and precise method to solve inverse problems.  We also explored the algebra behind finding the equation of the inverse relationship (given E, find D), and how one version of the formula can be manipulated to fall into the format y=[number]*x+[number].  With a computer doing the graphing for us, we could quickly manipulate those two numbers to see how changing those numbers changed the picture.

We then used that discussion to segue into a discussion on the formal concepts of “slope,” “y-intercept,” and “x-intercept.”  Andrew discussed a few misconceptions that he and Matt spotted on the posters from earlier.

We broke from that discussion to dive back into problem-solving, this time with the Burning the Candle problem.  Andrew rewrote the original version to remove scaffolding, which opened the question up to exploration and encouraged solvers to use methods that made the most sense to them.

After lunch, we talked more about funneling and focusing questions, with Fran handing out an example of 2 conversations between Ms. Jones and Li – one where Ms. Jones uses a funneling pattern and one where she uses a focusing pattern. Then at their tables, teachers talked about their pedagogy homework – what they learned about their own questioning patterns. We switched gears and spent the last 25 minutes of the pedagogy time to find problems in textbooks that support students’ “floating capacity” for doing/undoing processes, rate, proportional reasoning, linear relationships, using multiple representations.

After the pedagogy discussion, we returned to extract the “big ideas” from the Candle problem. The graph of height over time showed a line with negative slope, which was borne out by both the calculations involved and the geometry.  We saw how there were multiple choices of how to measure time: one can measure with x=0 representing midnight, 1am, 10:30pm (when the candle was list), or other times.  One could also measure x in hours or half-hours or minutes.  It was found that letting x representing “hours after midnight” (e.g., x=0 is midnight, x=1 is 1am) provided a convenient way to converting between values of x and clock times.

We then began exploring “What isn’t a linear relations?”  This is intended to help isolate important (and incidental) features of linear relations and proportions.

Pedagogy Homework

1.Read Principles to Action, the section titled “Access and Equity” (p. 59-69)

2. Finish the activity we started in class: Using your textbook or curriculum guide, identify content at your grade level that supports the “floating capacity” for doing/undoing processes, rate, proportional reasoning, linear relationships, using multiple representations.Transfer problems to a piece of paper.

Math Homework

1.  Return to the Kayaking Problem, Film Developing Problem (the original, not the variants), Currency Conversion without Exchange Fee, and Currency Conversion with Exchange Fee. Each involved comparing linear relationships. For each line involved describe each mathematical idea in terms of the original scenario. Specify units.

  • The slope of the line
  • The y-intercept of the line
  • The x-intercept of the line
  • The inverse relationship

2. Complete “What isn’t a linear relationship?”

Recap: Functions and Algebra Day 2 (July 28, 2015)

Fran began the morning by sharing a number of resources:

PMI Resources 1

  1. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School by Thomas P. Carpenter, Megan Loef Franke, and Linda Levi. (
  2. Classroom Discussions: Using Math Talk to Help Students Learn by Suzanne H. Chapin, Catherine O’Connor, and Nancy Canavan Anderson. (
  3. Extending Children’s Mathematics: Fractions and Decimals by Susan B. Empson and Linda Levi. (
  4. Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades by Susan Jo Russell, Deborah Schifter, and Virginia Bastable. (
  5. Putting the Practices into Action: Implementing the Common Core Standards for Mathematical Practice K-8 by Susan O’Connell and John SanGiovanni. (

We went deep into the homework on processes, comparing answers and pulling out some solid lessons on how to invert multi-step processes.   Along the way we also established the convention that a number-based process is most generally written by using a single-letter variable as the initial input. Jen also made the connection between finding inverses and fact families: effectively we have a two-member fact family  when we write a process and its inverse.

After the break we finished the discussion of the rice-and-water problem and how inverses play out there.  The original problem asked “Given R cups of rice, how many (W) cups of water do we need?”  The inverse problem asked “Given W cups of water, how many (R) cups of rice can that make?”  We ultimately came up with formulas for converting between R and W in terms of multiplying by a rate (the exact number we multiplied was 5/3 or 3/5 depending on whether we started knowing R or W).  By drawing both relations on the same axes (with different sets of labels), we drew an important geometric connection between a proportional relationship and its inverse.  Specifically, the graph of the inverse formula looks like the reflection across the line y=x (which can also be seen as the 45-degree line when two axes have the same scale, also the line through (1,1), (2,2), (3,3), etc).

The remainder of the math discussions  (morning and afternoon) centered around the problems in Unit FA-3 on Currency Conversion, both with and without an exchange rate.  This provides a vivid contrast between proportions (without exchange rate) and the more general idea of a linear relationship (with exchange rate.

The pedagogy discussions centered around “focusing versus funneling” question styles, and using student knowledge to guide your process.  After a discussion of what these terms mean, participants broke into pairs and engaged in role-playing exercises.  Partners recorded themselves as they took turns acting as students (who made errors) and their teacher.  The homework is to listen to one’s turn as teacher and analyze one’s teaching style.

Math Homework

Complete all 12 problems under the “Currency Conversion Problem with Exchange Rate”

Read pages FA3.9 to 10, on the terminology of linear relationships.

Pedagogy Homework

Listen to the audio-recording of your teaching episode. In your notebook, write down the questions that you asked.

In Principles to Actions, refer specifically to p. 37 (starting under Figure 14) and p. 39-40 Figure 16 “A comparison of questioning patterns…”

Examine the questions you used while playing the role of teacher and write responses to these prompts in your notebooks:

  • Were your questions funneling or focusing questions?
  • How might you change your questions now?
  • What types of questions might be more productive for eliciting student thinking? Write a few new questions based on the 4 questions types on page 36 and 37.

Also: Tomorrow (Wednesday) we will be getting into your textbooks. Please bring materials to look at!

Percolation assignment: When is it “ok” to use funneling questions?


Recap: Mathematics as a Second Language Day 4 (July 23, 2015)

We started the morning by making posters for the 8 “rules that expire” and having a short discussion about those rules. We handed out the article “13 Rules that Expire” from Teaching Children Mathematics. (Don’t forget that the “Nix the Tricks” download is available on our Google folder. You can also visit the website and submit your own ideas for “tricks” or “vocabulary” that can be problematic when teaching children.) Then teachers discussed their addition and subtraction with signed numbers word problems that they wrote for HW last night.

We began the math discussion by wrapping up a loose end from yesterday.  We used skip-counting and pattern recognition to justify the rules of sign for multiplication.

We then moved into Unit 5 on Division.  The first major idea is the dependence of division upon multiplication, like how subtraction is depended upon addition.  This has two ramifications: (1) a student’s mastery of multiplication has direct bearing on their potential to master division, and (2) we can justify properties of division by relating it to a corresponding multiplication problem.  We explored item (2) in depth as we saw why division by zero is problematic.  We skipped over the “Rules of Sign for Division” (page 5.10), but they are based directly on the rules of sign for multiplication.  We also had a side trip where we discussed how subtraction and division fail to be associative.

Just before lunch, Andrea shared a worksheet presenting several alternative algorithms for multiplication. We explored lattice multiplication and discussed how place value and the distributive property play out in that representation. You are encouraged to explore any of the other algorithms shared on that page. Bonus points if you can explain the Russian Peasant method!

We also spent some time exploring rate calculations and the adjective-noun theme for division.  The website that expands the adjective-noun theme throughout the K-8 curriculum is:  We will look at rates and unit conversion is more depth in Functions and Algebra.

After lunch, we had small group and then large group discussion of the pedagogy HW from last night. After a small-group discussion, we compiled teachers’ impressions of the beliefs about teaching and learning that they inferred for Ms. R and Ms. F from the reading. The teachers then talked in their groups about “things teachers can do to promote productive struggle in math class” and we compiled those as a whole group (we will continue to add to this list of strategies and then post the final list in the google folder at the end of next week).

Fran then shared her work from the Library Fines problem from yesterday – modeling productive struggle (she went down a number of dead end pathways, used a number of different representations – pictures, numbers, expressions).

Then we watched a video that showed a teacher using “My Favorite No” activity in her class as a way to promote productive struggle and risk-taking:

We spent the remainder of the afternoon exploring partitive and quotitive division problems.   The core difference is what information is given.  The total number of objects is the dividend (first number) in both cases, but in partitive division the divisor is the number of parts the total is split into, while in quotitive division the divisor is the size of each part.  We saw how quotitive division can also be called “measurement division” since it appears in conjunction with a nonstandard measure (e.g., “A pencil is 8 inches and a paperclip is 2 inches.  How many paper-clips-long is the pencil?”).  Beth provided another name for partitive division,”dividing to split,” and quotitive division, “dividing to fit.”

At the very end, Fran and Andrea discussed the Common Core Standards for Mathematical Practice.

Math Homework

  1. Finish the problems from “Exercises of Partitive and Quotitive Division of Whole Numbers”
  2. Problem 6 from “Practice Problems for Division,” page 5.16.  Actually making the paper rectangle is optional.
  3. The handshake problem.  Problem 5, page 2.11.

Pedagogy Homework

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. In your notebook, please write a response 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.


Recap: Mathematics as a Second Language, Day 3 (July 22, 2015)

We began the morning with teachers filling out the HW chart (Choosing Problems for HW Review on the HW Review Strategies list we’ll hand out at the end of the workshop). Teachers found a partner that they had not worked with already this week and compared/contrasted HW solutions.

After the homework review we moved into a discussion of signed numbers.  We explored how our models of addition and subtraction, particularly the chip model and number line models, extend to signed numbers.  We found that viewing subtraction as missing-addend addition (e.g., translating “a-b=[what?]” into “b + [what?] = a”) is very productive viewpoint when signed numbers are involved.  As a side diversion we discussed pitfalls of the word “cancellation,” and how that gives students trouble later in algebra.

Just before lunch, we watched a video of Carol Dweck (author of Mindsets) found at this URL:

After lunch, we talked about conceptual understanding and procedural fluency (last night’s reading assignment). Fran led the group in making an iceberg model for 3X4=12.



Then teachers worked in grade-band groups to create iceberg models for: comparing fractions, place value, addition to 20, and the coordinate plane. At the end of this session we watched another short video about praise and mindsets:

We then moved into Unit 4 and discussed multiplication and the distributive property.  We started with three models of multiplication: repeated addition, skip counting, and area.  We saw how repeated addition can give rise to each of the other models.  We also discussed how each model has its own limitations and strengths.  We then used those models to help us understand why the commutative and associative properties are true for multiplication, as well as why 1 and 0 have the multiplicative properties that they do.

We also discussed how the distributive property describes the interaction of multiplication and addition, and applied it to calculations.  We briefly connected “pulling out common factors” to reading the distributive property in reverse.  We also looked at how the distributive rule works more generally than what FOIL (“First Inner Outer Last”) can describe.

Math Homework

Read pages 3.28-3.32 on the opposite of a number and how it can be useful.  This likely matches with whatever you remember of how to add and subtract signed numbers.

Write word problems whose calculations involve the sums on page 3.24.  Try to do the same for the subtraction calculations on page 3.27.

Pedagogy Homework

Read Principles to Action, the section titled “Support Productive Struggle in Learning Mathematics” (p. 48-52).

In your notebook, write a response to this prompt:

  • Review the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles). What beliefs are evident in Ms. Flahive’s and Ms. Ramirez’s classrooms (see fig. 21 on page 51)? What impact do those beliefs have on students’ opportunities to grapple with the mathematical ideas and relationships in the problem?

Recap: Math as a Second Language, Day 2 (July 21, 2015)

We started Day 2 with Andrea leading a discussion of the burning questions from the day before, such as “How might ideas from Day 1, like understanding mathematical concepts as adjectives or nouns, play out in our classroom?” We then moved to a review of Day 1’s math homework. Teachers found a partner within their grade band (K-2, 3-4, or 5-8) to compare and contrast their solutions to the Film Problem. When we started the whole-group debriefing of the math homework, Fran asked the participants to show her, with thumbs up or thumbs down, if they were satisfied with their understandings of #1 and #2 based on their small group work. Seeing that everyone was satisfied, Fran stated the she was not going to do a whole-group review of the solution to the problems (no need since everyone was good with their solution).  Fran then asked if everyone was good with a solution to #3 and saw that there was some hesitation. A number of teachers shared their solutions to #3 (using different representations: tables of values, equations/expressions, graph). Fran then asked people who were not satisfied with their solutions to #4 and #5 to work further on those problems, using the new knowledge that they had built during this session.

Teachers then gathered in grade band groups and completed posters based on these prompts:

  • What ideas about expressions, equations, and multiple representations were involved with the hw problem?
  • How do similar ideas play out in your grade band?

We then moved onto small-group discussions of the pedagogy homework. Teachers shared curriculum materials in their groups and discussed the math tasks in their curriculum through a cognitive demand lens. Fran and Andrea ended this session with some whole-group discussion of cognitive demand, sharing information about the QUASAR project and how cognitive demand might be a useful lens to use with textbook adoption groups.

We spent the rest of the morning discussion how counters can inform our understanding of numbers and addition.  We used geometry to gain a new view of even and odd numbers in terms of stacks of 2, and extended those ideas to looking at remainders when numbers are divided by numbers other than 2.

After lunch we tackled the “100 days of Christmas” problem, which led to a lot of fruitful discussions.   There is a summary of many of the arguments made in class (and others) at

We next began our discussion of subtraction by basing it on the missing-addend form of addition.  Nola shared how students at the end of first grade are already seeing addition with missing addends, which sets up subtraction.  We then moved into multiple models of subtraction, both in terms of counters and a number line, eventually settling on distance as a view that will be most flexible.

We also began to explore what the adjective-noun theme has to say about subtraction.  The short version is that it works the same as addition, which isn’t surprising considering the deep connection between subtract and addition..  We also considered alternate algorithms for multi-digit subtraction as a test of our understanding of how place value interacts with subtraction.

Math Homework

“Karl Friedrich Gauss Library Overdue Fees”  (Treat the variation as a challenge)

“Problems on Subtraction” (pp. 3.15-3.17) Problem 7

Pedagogical Homework

Read Principles to Action: Section titled “Build Procedural Fluency from Conceptual Understanding” (p. 42-48).

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

In your notebook:

  • Identify a procedure or skill that you consider essential for students at your grade level to learn. Describe the conceptual understandings that support students’ learning of the procedure or skill.
  • Write questions you have based on the 2 readings.

Odds and Ends

The group that wrote the Common Core Standards are also writing documents detailing how certain ideas progress across grade levels.  These are currently in draft form, but are usable. Drafts are available at  A table summarizing the various combinations of given and missing terms in an addition or subtraction problem appears on page 7 of the K-5 Progression on Counting and Cardinality and Operations and Algebraic Thinking.

Andrew uses the following to help him research origins of terms and notation