Monthly Archives: July 2016

Day 4 of Functions & Algebra (7/28/16)

Today we began with a discussion about the symbolic rules that describe the tortoise and hare problem. Fran emphasized that writing symbolic rules can be supported by working through the reasoning in other representations. Andrea shared some data about young children’s misunderstandings of the meaning of the equal sign and practices to avoid. We also watched a video of a young child using relational thinking to solve an open number problem.

We worked on the candle burning problem, which illustrates how a negative rate of change effects a linear relationship. After debriefing, Andrew talked through consequences of linearity.  We ended the morning with analyzing 12 situations to determine whether they could be solved via a linear relationship, a proportion, or neither.

After lunch we wrapped up some of our big pedagogy ideas by reflecting on last night’s messages, generating a list of things that need to happen in order to implement the strategies. We discussed more about the Standards for Mathematical Practice , focusing especially on supporting students to persevere.

We constructed a big iceberg wall together, representing ideas at each grade level that 20160729_104509contribute floating capacity to the concept of linear relationships.


We closed with a debriefing of the 12 situations on “What isn’t a linear relationship?” and highlighted the important features that distinguish them from each other.

See you tomorrow for our last day together!


Pedagogy: Read the section on Professionalism in Principles to Actions (pp. 99- 108).

Math:  None.

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.




Day 2 of Functions & Algebra (7/26/16)

We began by reviewing last night’s math homework using the “Jigsaw” strategy.

Then we moved on in our Functions and Algebra notes, and we learned about Dan Meyer’s 3-act problems.  We looked at what makes proportional relationships special, and then pushed ourselves to solve problems based on proportions.

After lunch we discussed our responses to last night’s readings from Principles to Actions about questioning and eliciting student thinking. We used the reading to analyze the questioning practices of an 8th grade teacher leading students in the “Water Tank Task.” We both watched a video of her teaching and read a transcript episode.

We started our discussion of unit FA-3: Linear Relationships, but did not get past the first problem on currency conversion.  We will pick up there Wednesday morning.

Math Homework:

  1.  Create a “sequel” to one of the problems (1-5) on page FA2.12 that pushes the solver to use a systematic or efficient strategy to solve it.
  2. Revisit the milk problem on page FA2.12.

Pedagogy Homework:

  1. Look through your curriculum materials and identify a lesson that you might teach in the first few weeks of school. Refer to the picture below as a reference to find two “traditional problems” and rephrase them to make them “rich problems.” (Figure is from “Putting the Practices into Action” by O’Connell & SanGiovanni, published in 2013 by Heinemann).
  2. Also look through your curriculum materials and identify at least one lesson or topic that you teach that helps students build “floating capacity” for an idea we’ve discussed in the “Functions and Algebra” course.
  3. Read the “Upside-down teaching” article and think about how this connects to some of the time constraints we’ve discussed in class.



Day 1 of Functions & Algebra (7/25/16)

We are back for Week 2!

Walk the Graph

We began the morning with walking the graph.

FA-1: Processes

Andrew led discussion on the informal concept of “processes.”  This is like input-output machines that many teachers are already familiar with.  Our discussion took us many places, but some of the highlights were:

  1. Some processes are invertible, that is, they can be undone to return to the original input for any input.  Inverse operations generalize the idea of how addition and subtraction are inverses of each other, and how multiplication and division are inverses of each other.
  2. Some processes are not invertible, such as multiplication by 0.  Knowing that a number produced 0 as the output is not enough to know what the original number is.
  3. We can break more complicated procedures into smaller processes.
  4. When chaining together procedures, the order matters.  For example, “multiply by 2 then add 3” is different from “add 3 then multiply by 2.”
  5. We indicate the order of procedures by using parentheses in an algebraic expression.
  6. Two procedures are equivalent if they always produce the same output given the same input.  For example, “add 3 then multiply by 2” is a different process than “multiply by 2 then add six,” but they are equivalent in that they always produce the same output given the same input. In other words, they represent different journeys to the same destination.

Role-playing helping a student

We divided into pairs and took turns role-playing a teacher helping a student with a mathematical misunderstanding. We audio-recorded these so that we can use them for tonight’s homework.

FA-2: Proportions

Matt led discussion on Proportions.  We explored the relationship between the number of feet and the corresponding number of inches, and contrasted it with a length and the area of a square with that side length.  We will conclude the discussion and work on related problems Tuesday.



Complete problems 1, 2, 3 on page FA1.9–10.


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 responses 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?

Day 5 of Math as a Second Language (7/22/2016)

Homework Review

We began by reviewing the homework.  Different tables discussed different problems, and we had some good back-and-forth about factoring and dividing to one.  Fran highlighted how a bit part of algebra success is recognizing certain features such as common factors.


Then we moved our chairs so that we could sit in a big circle and discuss the norms that we–as a group– have established this week. Here’s a list of what we talked about:

  • Making mistakes is okay.
  • Five minutes means five minutes.
  • Working in a group helps learn math.
  • Be a good group member.
  • Ask questions when you don’t understand something.
  • Create a safe environment for students.
  • Be an active participant.
  • Multiple approaches to solving math problems are valued.
  • Explain your thinking.
  • Teachers said “explain your answer” whether or not the answer is right or wrong.
  • Try your best
  • Establish routines for “housekeeping” (e.g., getting quiet)

Group Picture

We got a group picture, which will be posted soon.

Fraction Addition and Subtraction

We began our discussion about fraction addition and subtraction, starting with length-based models and how that can support what sums of (positive) fractions should be.  We repeated (frequently) about how our different models of fractions can grant different insights into the arithmetic.  For example, length-based models demonstrate what the sum should be, and finding an appropriate way to measure that length brings about a common denominator.  Fraction models based on amount-of-pieces (e.g., slices of pizza) can present a hurdle (which can be overcome) when the sum is more than one.  Viewing fractions as proportions (i.e. seeing 1/2 as one out of every two) is ill-suited for adding fractions.

Fraction Multiplication

We next moved into fraction multiplication by exploring 4 interpretations of (2/3)×(4/5):

  1. Add up (2/3) copies of (4/5)
  2. Find the area of a rectangle with length (2/3) and width (4/5)
  3. Find (2/3) of (4/5)
  4. A ribbon is 4/5 feet long, and is cut into 3 equal-length pieces.  How long would 2 of them be?

We found that #1, based on repeated addition, is ill-suited for multiplication when neither is a whole number.  Option #3 isn’t much better, at least as far as intuition-building.  Models 2 and 4 were most convincing.  We spent the rest of the morning discussing our area models for #2, progressing to other examples, one where one factor is greater than one, and another where the product is greater than one.  The area model reinforced how the numerator of a fraction is an amount of pieces while the denominator describes how small the pieces are.


Fraction Division

We wrapped up the discussion of fraction multiplication by briefly exploring interpretation #4 above and cutting a fractional length into equal pieces.  That problem reinforced the “2/3 is two pieces of size 1/3” viewpoint.

We also briefly discussed why mixed number multiplication can be problematic.

After eating lunch together, we resumed our discussion of fraction division. We drew pictures, compared models, and learned about the common denominator method!

“I’m Stuck!”

We began the pedagogy part of the day by compiling suggestions for a “I’m Stuck” poster:

  • When in doubt, draw it out
  • Try for 10 minutes and then walk away
  • Read the room
  • Ask a partner
  • Explain what you do know
  • Make a table
  • What do you need to know
  • Decompose/recompose or rename numbers
  • Use a manipulative or other kind of manipulative
  • Use a model (tape model, number line, array)
  • Restart the problem in a new way (pick a new pathway)
  • Reread problem (out loud)
  • Rewrite problem
  • Ask the teacher (a specific question)

Planning Lessons that support our goals

We then picked up our work with our grade-level colleagues and looked for more high-level tasks and plan for supporting productive struggle.

We concluded our day by reflecting on our take-away messages and saying good-bye to some of our colleagues. It’s been a busy but fun week!


Enjoy the weekend, and come back refreshed on Monday morning.

(For the record, breakfast will not be provided Monday.)

Day 4 of Math as a Second Language (7/21/2016)

Rules that Expire

We began the morning with “Rules that Expire” – we had 10 rules and groups of teachers made a presentation about their particular rule that expires. All of the rules came from an article published in Teaching Children Mathematics titled “13 Rules that Expire,” which we gave to each teacher.

Rules of Sign

We revisited the why of (Negative)x(Negative)=(Positive).  Andrew shared two arguments (one from Matt), Fran shared a third, and Hope opened the discussion to introduce a fourth.  These four arguments were all based on the principle that arithmetic needs to be consistent, rather than cute analogies or contrived scenarios.

Unit 5: Division as Unmultiplication

We next moved into discussing division, starting with “walking back” the concept of how division is treated in fifth grade and then working grade-by-grade back to Kindergarten.  We found how the seeds of division are planted as early as Kindergarten.   We based our understanding of division on “unmultiplication,” specifically how any division calculation can be seen as filling in a missing factor for a multiplication sentence.  This perspective allows us to leverage our models of multiplication to inform how we view division.


Pedagogy: Productive Struggle Discussion

We began by compiling a list about beliefs about teaching/learning mathematics we interpreted from the description of Ms. Flahive’s and Ms. Rameriz’s classrooms (in the P to A reading). We then compiled a list of strategies a teacher can employ to support productive struggle, including:

  • create an atmosphere where it’s ok to make mistakes (and can help get the parents on board with this idea, too)
  • create and enforce effective boundaries and constraints with time
  • model their own thinking processes and their own math struggles
  • provide a mix of high-demand and low-demand tasks
  • provide a variety of tools and materials so that students can solve problems in different ways
  • allow students to communicate with each other and with you
  • require an explanation and not just an answer (students should expect to do this!)
  • prioritize hard work over natural ability/intelligence (growth vs. fixed mindset)
  • provide suggestions—not answers—to help students keep working productively (facilitating learning)
  • provide resources (like a poster) showing suggestions for getting “unstuck”
  • validate that you understand this is challenging
  • keep an open mind—be open to learning and trying new things
  • provide immediate feedback based on effort—praise effort (even if it’s only 1 part of a problem—find what they did do correctly)

We then split up by grade level and found (or created) a task that requires a high level of cognitive demand and began to plan for implementation, with a specific focus on supporting productive struggle.

We ended pedagogy time today by watching a video about mindsets:

Rates, and the Adjective-Noun theme for Multiplication and Division

We discussed how multiplication can be seen in conjunction with rates, such as how paying 2 dollars/hour for 3 hours of parking costs $6 since 2×3=6.  Going the opposite direction, paying $6 for 3 hours of parking means that the rate was 2 dollars/hour since 6÷3=2.

Rates play a fundamental role in grades 6, 7, and beyond.

Fraction Fundamentals

We next moved into a discussion of how participants teach fractions and how they themselves understand them.  We found myriad ways to represent a fraction as simple as 2/3 (see Andrew’s summary image below), but also found representations like “2 out of 3” to be limited when moving to improper fractions like 5/3.  Other models, and especially the number line model, work just as well for proper fractions as for improper fractions.


We closed with a discussion of equivalent fractions and the idea of “dividing to one” to justify when two fractions have equal value.  In short, we used area models to justify why 2/3 could be changed to 8/12 by cutting the rectangle to even smaller pieces and using more of them; then we read that equivalence backward to justify why 8/12 could be simplified to 2/3.  (We also discussed the pitfalls of using words like “reducing fractions” and “cancellation.)


Math Content:

Page 5.16, problem 6.
Page 6.18, problem 1&2.


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. Please respond 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.

Day 3 of Math as a Second Language (7/20/2016)

In order to discuss the homework from last night (the Library Fines problem and the temperature problems), we used the homework review strategy of “Using a Chart to Choose Items for Discussion.”

Then we continued yesterday’s discussion on addition and subtraction of signed numbers, and we began our consideration of multiplication. What IS multiplication, anyway? Turns out: it is repeated addition, but it’s a lot of other things too!

We were able to enjoy some beautiful weather during our lunch break. Upon return, we discussed conceptual understanding and procedural fluency and how these two dimensions of mathematical proficiency can complement one another.

We applied ideas from the “iceberg” article to construct our own gallery of posters. These posters let us illustrate the conceptual understandings that lend “floating capacity” to the “icebergs” of mathematical procedures. The posters look great and contain many important ideas. See pictures below.

IMG_1187 IMG_1186 IMG_1185 IMG_1184 IMG_1183

Finally we extended our discussion of multiplication to consider how the distributive property lays out the relationship between addition and multiplication. We also saw how factoring common terms, such as (6+15)=3*(2+5), is only the distributive rule in reverse.

Homework for tomorrow (Thursday):

Math HW:
Problem 9 on page 4.7 (area of a staircase).  (The area is not 32).  If problem 9 was easy for you, do 10.

Problem 7 on page 4.23 (multiplying by 9 trick).

Pedagogy HW:

Read Principles to Actions, 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?

Day 2 of Math as a Second Language (7/19/2016)

Today was another great, math-filled day!

We began the day by “circling up” and introducing ourselves to the whole groups. We each shared something fun that we will or have done this summer. We’ve got beach lovers, roller-coaster riders, Disney-goers, and animal caretakers. Yay for summer!

We were paired up with a colleague from a different district and we compared our work on the homework problems–variations on the “kayak problem” and the “tennis trophy problem” about evens and odds. We discussed the variety of valid interpretations for the problems based on the wording and based on what we know from the “real world” about kayaking and tennis tournaments.

Then Andrew launched the “Days of Christmas problem” and we set off on problem-solving. Triangular numbers! Square numbers! Gauss! Searching for patterns in tables and numbers! Making triangles and squares with counters! Generating expressions and conjectures!

Pam shared a great resource: MDC (mathematics design collaborative) through the SAS website. (for members only, but it’s free, so join!)

After breaking for lunch, we watched a video of 3rd graders solving the handshake problem. We made posters presenting how ideas related to expressions and equations play out at each of our grade levels, and then we displayed these posters in order to look for themes and alignment. (pictures below)

IMG_1182 IMG_1181 IMG_1180 IMG_1179 IMG_1178 IMG_1177 IMG_1176 IMG_1175


Fran and Andrea led a discussion about low-demand tasks and high-demand tasks and the importance of providing balanced opportunities for students to practice all types of mathematical thinking.

We concluded with Andrew’s presentation of Unit 3: Subtraction as the Art of Unadding. (Note: Spell check does not like the word ‘unadding.”)

Homework for tomorrow (Wednesday):

Math HW:

Work on the Library Problem

Work on the problems about temperatures (p. 3.19 #15 and #16)

Pedagogy HW:

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 about the readings.

Day 1 of Math as a Second Language (7/18/2016)

This has been an  exciting first day!

We started with the usual introductions, where you were greeted by Andrew Baxter, George Andrews, Yuxi Zheng, Matt Katz, Andrea McCloskey, and Fran Arbaugh.   We talked about expectations and our objectives for this week, and then dove into the Kayak Problem.

In discussing our solutions to the Kayak Problem we discovered a few things about mathematics and mathematics teaching:

  • There is more than one way to solve a problem, although often the ways to solve are related.  Here we saw a list of +2s and +4s to represent the hours, a organized chart, an algebraic formula, and a graphical representation.
  • There are advantages and disadvantages of these different methods, as some solutions make certain subtleties and patterns clearer than others.  Ultimately students should develop proficiency in the efficient methods, but there is danger in knowing only those methods at the exclusion of mathematical understanding.

After a break, the participants took a pre-assessment.  This was followed by a lunch hour.

After lunch Fran led a discussion about the reading you did before coming to UP: “Smarter Than We Think: Helping Students Grow Their Minds” by Cathy Seeley. She then shared three frameworks that ground our pedagogy work for PMI:

  1. What does it mean to know mathematics? We will draw on work presented in Adding it Up: Helping Children Learn Mathematics, which presents a definition of “mathematical proficiency” as being comprised of the following five strands:

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.

Adding it Up can be downloaded for free at:

2.  What kind of activity do students need to engage in to become mathematically proficient?  We will draw on CCSS-M Standards for Mathematical Practice (found on page 8 of Principles to Action).

3. What kind of teaching practices support students to engage in the SMPs and become mathematically proficient? We will draw on the content of Principles to Action.

Next Matt kicked off Unit 2 on the Adjective-Noun Theme and Addition.  We saw how in arithmetic situations, numbers are adjectives and we often omit (or forget) about the nouns.   This can shed light on why our standard algorithm for addition works as it does, as well as give insight into why fractions need common denominators when adding and subtracting.

Last Andrew discussed models for numbers, their relative strengths and weaknesses, and how those models for numbers yield models for addition.  We started on exploring even and off numbers, but we will pick up with that more tomorrow

HOMEWORK: One clarification: homework is not collected and graded or for anything other than  deepening and solidifying your own learning. Homework problems are discussed the following day.


Page 1.5, Variations on the Kayak Problem. Solve problems 3 and 4. One you solved it one way, seek out another way to solve it.

Page 2.11, number 7. Woodchuck Valley Tennis Association Trophies. Make an attempt, although we will look at this in class in the morning.


Read Principles to Action: Sections titled Progress and Change; Effective Teaching & Learning (Pages 1-12) and 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), 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?