Algebraic Thinking and Proportional Reasoning, Day 4 (7/20/17)

We began the day by discussing the reading on linear relationships. Beth emphasized the idea that the proportional relationships we had been analyzing on Tuesday are really a special case of linear relationships: All proportional relationships are linear, but not all linear relationships are proportional. The proportional relationships are linear relationships with a zero intercept – in other words, the graph of the relationship passes through the origin.

Next, we dived into the “Burning the Candle” problem. Everyone was able to find numerical answers quickly, but it was harder to come up with the formula that describes this situation.

Resources discussed in the pedagogy discussion:

Doc Jul 20, 2017, 11_42-1xkyp0o

At the end of lunch, we took a group photo! Here it is:

In the afternoon, we examined a large set of relationships and decided whether or not they were linear. If they were linear, we decided whether or not they were proportional. We summarized by describing how we could recognize relationships:

For a Proportional relationship:

  • The graph must be a line and must pass through (0,0).
  • There must be a constant rate of change.
  • The two quantities must have a constant ratio

For a linear relationship:

  • The graph must be a line, but it does not need to pass through (0,0)
  • There must be a constant rate of change.
  • The two quantities do not necessarily have a constant ratio

For a relationship that is not linear:

  • The graph doesn’t make a line
  • There can be an exponent other than 1 or 0 (you can think about this one if you’re comfortable with the idea of exponents)

Notes from the afternoon pedagogy discussion:

A rich task…

  • More than one way to approach the problem
  • Opportunity for revision
  • Not focused exclusively on procedures
  • Connects to real life
  • Independent/cooperative learning
  • Analyze others’ solutions
  • Encourages math discourse
  • Productive struggle
  • Builds on prior knowledge
  • Requires adequate time
  • Engaging/Raises curiosity
  • Address misconceptions
  • Students creating their own problem.

Resources for Rich Tasks

    • NCTM publications: Teaching Children’s Mathematics Journal
    • NCTM Illumination:
    • Illustrative Mathematics Project:
    • Mathematics Assessment Project
    • Dan Meyer’s blog. Eg: Nana’s Chocolate Milk:
    • Yummy Math:
    • NextLesson :
    • LearnZillion:

Algebraic Thinking and Proportional Reasoning, Day 3 (7/19/17)

We started the day by using the Jigsaw strategy to review the homework. Each table worked together to be an expert in one part of the homework problems, then we shifted around so that each group had an expert in one problem present, and the groups shared.

Next, participants worked through the Currency Exchange problem. This is another proportional relationship, but we’re extending our thinking to the more general class of linear relationships now. One particular challenge was writing the equation relating D (price in dollars) to E (price in Euros). Many people make the “reversal error,” in which they (incorrectly) translate the statement “$1.00 is worth €0.90” to “D = 0.9 E”.

After lunch, we did the Currency Exchange with a fee. We described how to represent this situation with an equation, or with a graph. We noticed that when we graphed the original currency exchange equation on the same axes as the currency exchange with a fee, the lines were parallel. We compared the equations and related this to the slope of the graphs.

Some notes from the Pedagogy discussions today:

Practice Math  Talk moves Examples
Make sense of problems and persevere in solving them.


Wait time If you’re finished show another way to solve and compare your answers


Look for and express regularity in repeated reasoning.


Repeating Using a prior explanation from another student to solve a similar problem with different numbers

Extending the structure of a problem (solving an equation—word problem)

Reasoning Can you model your solution in a different way?
Model with mathematics Turn and Talk Modeling equivalent fractions with manipulatives (each group uses different manipulatives)
Look for and make use of structure.


Reasoning How is your comparison different from your partner’s?  Can you explain?
Construct viable arguments and critique the reasoning of others.


Reasoning Did anyone use a different strategy to solve the problem or same strategy with a different answer?
Attend to precision.


Revising Would you like to revise your writing/thinking using specific math vocabulary
Adding On Clarify or add more detail

Convert units—explain or rounding in context to the problem


Math: Read Block 3, p. 11 – 12.

Pedagogy: Read Message 12: “Upside Down Teaching”. Think about a task at your grade level that you can “turn upside down”

If you have extra time: For pleasure read “13 Rules that Expire” article.

Bring your curricular materials tomorrow!

Algebraic Thinking and Proportional Reasoning, Day 2 (7/18/17)

Today, we spent a LOT of time on proportions! Pretty appropriate, given how important these ideas are. We started by discussing what is or is not a proportional relationship. If two quantities are proportional, their ratio (one divided by the other) is constant. So for instance, the ratio might be 1/3 or 25/7. A ratio of 1/x is not proportional because it varies – it depends on the value of x. Even if you see a clear pattern between quantities, the relationship is not necessarily proportional.

We represented proportional relationships using algebra, and also using tape diagrams. In order to make sense of the algebra, we decided to return to “Math as a Second Language”, block 3, pages 31 – 35. This helped us to think about dividing with fractions, which was useful for the ratios!

Notes from the pedagogy discussion:

Characteristics of Examples of Funneling pattern of questions Characteristics of Examples of Focusing pattern of questions
-aims to move student toward math goal but discards students’ thinking







·      Do you see a pattern?

·      Are your columns lined up correctly?

·      Have you tried x strategy?

·      Is that really what that tells us?

·      Is she right?

·      What’s the answer?

·      What is a(n) … ?

·      What does it tell us?

-elicits & builds on students’ thinking, aims for students to communicate, interact, reflect.

– Require careful listening/monitoring/interpreting, allowing wait time, anticipating possible students’ approaches



·      Explain your thinking.

·      (Can) someone else explain this in their own words?

·      How did you organize your work?

·      Justify!

·      Why?

·      Please show a different strategy.

·      What if … ?

·      Please find a counterexample.

·      What might … ?

·      How does the …help us?

·      How do you know that?

·      Can someone elaborate on…?


Note: We decided to rephrase questions here so that they do not lead to a Yes/No answer.

How to pose purposeful questions:

  1. Knowing math content: learning trajectories, how students develop knowledge, floating capacity for that concept.
  2. Knowing your students, what they know, where they struggle & make mistakes.
  3. Establish explicit learning goals and plan questions accordingly
  4. Noticing & Gather evidence about students understanding and respond productively.




  1. Complete Block 2, page 8.
  2. Complete Block 2, page 11, #1.


Last names beginning A – M: Read “Facilitate Meaningful Math Discourse” from P to A (pp. 29-35) (Last names A-M)

List and explain 3-4 different strategies you use/might use to facilitate meaningful math discourse in your classroom.

Last names beginning A – M: Read “a Kid Can Say” article (Last names N-Z).

List and explain 3-4 strategies you use/might use to engage your students in “real learning”?

Algebraic Thinking and Proportional Reasoning, Day 1 (7/17/17)

Welcome back for week two! Today we welcomed several friends returning from last week, as well as several participants who completed Week 1 last summer, and are now back for week 2.

We began the morning with the “Walk the Graph” activity. Beth brought us to the physics lab, where motion detectors and computers were set up.  We touched on a lot of ideas that will show again this week, although in less formality.  During discussion we saw how distance from the detector, speed of motion, and direction of motion could all be visible in the graph. We also discussed a few graphs that could not be generated at all – they required someone to be in multiple positions at the same time.

Next, we moved back to the Math room and discussed Block 0, an overview of Week 2. A key point to remember is that the material we’re covering in week 2 is probably not directly applicable to what you teach in your classroom. These activities will really challenge you – don’t be afraid to ask for support from your peers or from the discussion leaders when you need it. These tasks are definitely not meant to be used (in their current form) with your students!

Next we tackled the “Sliding Snail” problem. Many people incorrectly concluded that it would take 24 hours for the snail to reach the top of the ramp (12 feet). However, after some discussion, most people realized that the snail would actually pass 12 feet earlier than 24 hours – because as soon as he gets to the top, he won’t slide down anymore. Participants shared their solutions, including tables of values and graphs. Lin showed how to use technology (in this case, the app Geogebra – ) to show a graph.

Next we moved into thinking about even numbers and odd numbers. We extended the thinking we used with counters to think about divisibility and remainders.

After lunch, we considered processes – We discussed the idea that some processes are interchangeable (add 6 to the input vs. add the input to 6) while others are not (subtract 6 from the input vs. subtract the input from 6).


Math Homework:

Try to find the process that makes Mathemagic Trick #2 work. (See Block 1 page 12).

Hint: For any number that is a multiple of 99, the sum of the digits will equal 18.

Pedagogy Homework:

Read Principles to Action: “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. If Nursen came to your classroom during math time, would she see you using a funnelling pattern of questioning or a focusing pattern of questioning?
  3. Suppose a student presented you with the following piece of work. What questions could you ask that would elicit the student’s thinking?

Screen Shot 2016-08-03 at 2.22.52 PM

Math as a Second Language, Day 5 (7/14/17)

We began today by looking at how the Standard Algorithm for multi-digit multiplication rests on a foundation of the distributive property of multiplication. We also looked at how the Adjective-Noun theme applies in multiplication: when you’re multiplying, you multiply the adjectives and multiply the nouns. This will have implications for how we treat the multiplication of fractions.

Then we thought about multiplying fractions, using an area model. When we practiced, we realized that multiplying whole numbers by fractions can get a bit confusing. We need to keep careful track of what constitutes our “whole” in this case. Using the number line can help here, too.

After morning break, Nursen led a discussion of productive struggle. She showed the “My favorite no” video. During the discussion, the group generated the following:

Beliefs about Teaching and Learning Mathematics Beliefs about Teaching and Learning Mathematics
Ms. Flahive (Ms. F) Ms. Ramirez

Closed mindset

Friend to students

Makes math easier (for her & kids)

Gives explicit strategy/instruction

Delivering information

Concerned with product not process

Keeper of the information

Control—not sharing authority

Avoids frustration

Removes challenges

Task requires reasoning & problem solving


Facilitator (teacher as)

Encouraging work through challenges

Wants students to use multiple approaches

Building on prior knowledge

Poses purposeful questions

Class discussions/prioritize mathematical discourse

Supports productive struggle

Utilizes strengths of students

Task requires reasoning & problem solving

Access to tools (materials, manipulatives)

Valued the process

Praises effort


To support productive struggle in learning mathematics, teachers can:

  • create a learning atmosphere where it’s ok to make mistakes
  • monitor carefully the students’ progress
  • model their own thinking processes and their own math struggles
  • provide a variety of tools and materials
  • invite multiple solutions
  • promote student-student interaction: evaluating and critiquing others’ approaches
  • require an explanation, justification. Use “Convince me” phrase.
  • Focus on the process not the product
  • prioritize hard work over natural ability/intelligence (growth vs. fixed mindset)
  • provide suggestions to help students make progress toward the learning goal.
  • Model and Provide suggestions to students in terms of how they should communicate
  • Establish clear math goals.
  • Purposeful pairing, grouping
  • provide immediate and helpful feedback based on effort—praise effort
  • honor students’ thinking.
  • Select activities/tasks that aligns with the learning goals and provide opportunities for productive struggle

When I get stuck, I can…

  • Use something I already know (from the problem; strategies / methods)
  • Draw a picture
  • Refer to my notes / textbook
  • Use my tools / manipulatives
  • Talk to my neighbor
  • Start again / try other methods
  • Ask the teacher (Am I on the right track?)
  • Explain my thinking and ask for “What is it that I am missing?”
  • Look for key words, hints.
  • Break the problem down into smaller steps.
  • Look for similar problems that I have done in the past.
  • Use tables, graphs, charts
  • Make mistakes 🙂

After lunch, we returned for the final math block. Beth led a discussion on division and the differences between Partitive Division or Measurement Division (she calls these “Dividing to split” and “Dividing to fit”). We pointed out how Measurement division is supported by a foundation of measurement using non-standard units that kids are doing as early as Kindergarten (“How many paper clips fit in this marker?”).

(Note: Nursen pointed me at a video that goes through how Division ideas progress through the grade levels. You can see it here:


Pedagogy Homework for Tuesday:

Read Principles to Action: “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. If Nursen came to your classroom during math time, would she see you using a funnelling pattern of questioning or a focusing pattern of questioning?
  3. Suppose a student presented you with the following piece of work. What questions could you ask that would elicit the student’s thinking?

Screen Shot 2016-08-03 at 2.22.52 PM

Math as a Second Language, Day 4 (7/13/17)

This morning, we began by reviewing the “100 Days of Christmas” and the Handshake problems. We connected each of them to the triangular number problem from the previous day.

Then we dove into more math – formally introducing the idea of multiplication. We discussed the power of the area model for thinking about multiplication of fractions. We related the stacks of pennies from the Piggy Bank problem to the one-to-many correspondence model.

We examined various properties of multiplication: the commutative property and the associative property and justified them using the models of multiplication.

[then, Nursen led a pedagogy discussion]

After lunch, we returned to the properties of multiplication. We dug into the distributive property, and how this can be justified using the area model, and how it supports the algebra skills that students will need in later math courses. Finally, we examined the multiplicative property of one, the multiplicative property of zero, and the multiplicative property of -1.

[then, Nursen led a pedagogy discussion. Participants were grouped by grade bands, and made “icebergs” for a procedure or skill that students develop in their grade band.]


Math Homework:

Complete Block 3, p. 12, and the new page on “Adjective-Noun theme for multiplication.”

Pedagogy Homework:

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?
Find a task/activity from your curricular material(s) that supports productive struggle.

Math as a Second Language Day 3 (7/12/17)

We began the day by reviewing some of the math homework from yesterday. Everyone shared the problems they had written by writing them on post-it notes and then sticking them to posters that had been put up around the room. It became clear that the most common subtraction problems involved comparing, with an unknown difference (Lucy has A apples. Julie has C apples. How many more apples does Julie have than Lucy?). Participants suggested that this might be because this is a fairly standard basic subtraction. In contrast, very few people wrote problems involving both addends being unknown. We discussed the ways in which this also is a useful skill. Participants noted how they need to check themselves and make sure that they are writing problems that demonstrate the full spectrum of subtraction. Elyse suggested that with advanced students, she might give them the reference sheet on the spectrum of subtraction, and ask the to write their own problems.

After the homework review, we did the sheet on comparing fractions. Participants grappled with relating the fractions to benchmark numbers, and thinking about whether one piece was being added or taken away, and also considering the size of the pieces.

Next, we started thinking about signed numbers.We discussed how to model positive and negative numbers in terms of the number line, and also using chips of different colors to represent positive/negative numbers. We used these models to think about adding and subtracting signed numbers. To make sense of subtracting signed numbers, we re-wrote the subtraction problems as addition problems.

After lunch, Nursen led a pedagogy discussion.



Math Homework:

Block 2, page 36, #5 and #6. (Do not spend more than 30 minutes on these two problems together. If you haven’t gotten an answer by the end of 30 minutes, write down what you know, and some questions that would help you to continue before setting it aside.)

Pedagogy Homework:

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 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.
  • List 2-4 ways that the math work you’ve been doing in PMI connect to these two readings.

Math as a Second Language Day 2 (7/11/17)

Welcome back for day 2! We started the morning by reviewing some comments from yesterday’s evaluations. We want to make sure that everyone feels comfortable letting us know if you can’t hear what we’re saying, or if you’re unclear what you are supposed to be doing at any given time.

Also, a note about the curriculum: Many of the pages are marked “Reference.” These are designed for you to read and refer to outside of class time. In most cases, we make sure that the key ideas for the “Reference” pages emerge in the discussion in class, but we don’t spend time actually going through the reference pages during class.

For our first activity of the morning, we reviewed the homework, “Rules that Expire.” Each table created a poster for one of the rules discussing when it is useful and when it fails.

Andrew shared a new worksheet he had created to help share different ways we use addition. Notice how addition can be cued by words that we more commonly associate with subtraction! (For instance, “less than” can be used to create an addition problem.) We discussed how some of these problems are better modeled by adding amounts, while others can be modeled as lengths, or as positions on the number line.

Next, we spent some time on the Commutative and Associative properties of addition. Groups discussed how to use counters, or the number line, to help their students understand these properties. One group shared a method using a post-it note to represent grouping counters in parentheses, and then demonstrated how they could stack the counters to show their students the associative property.

Next, we discussed the idea of “subtraction as take-away”. Many flavors of subtraction don’t really feel like they involve taking something away! Andrew pointed out that this is one of the reasons that research shows that subtraction is the hardest operation for students to understand.

After lunch, we were introduced to the Adjective-Noun theme. Numbers are adjectives modifying some noun. We discussed how important it in in addition and subtraction to make sure that the nouns match. You add the adjectives and keep the nouns. The adjective-noun theme can help to motivate the idea of place values, and the addition and subtraction of fractions.

Before the afternoon break, we watched a video from Veritasium about dual-process theory:

Math Homework:

  1. Read Block 2, pp 7 -8. Generate 3 math problems (with numbers appropriate to your grade) to represent 3 different boxes on the grid.
  2. Complete Block 2, page 9, #2.

Pedagogy Homework:

  1. Read Cathy Seeley’s chapter: Message #1: Smarter Than We Think: Helping Students Grow Their Minds.
  2. In your notebooks, write a response to these prompts:
  • What struck you about the content of the “Smarter Than We Think” message?
  • What are a few connections that you see between this reading and the reading from Principles to Actions from last night?


Math as a Second Language: Day 1 (7/10/17)

We began the morning by introducing ourselves. The people leading the workshop at PSU-GA this year are Dr. Beth Lindsey, Associate Professor of Physics, and Dr. Kuei-Nuan Lin, Assistant Professor of Mathematics. We also have a student, Ashley Brandon, who is helping us out this week. We received a warm welcome from Dr. Jackie Edmondson, the campus Chancellor.

After the welcome, we dove into some math! Participants worked on the “Kayak Problem” on their own, and then discussed the problem in their small groups. Many people had used a table or chart to find the times at which the two rental places cost the same amount. Shanna shared how she had used a multiplication approach to solve part A. Katie took the multiplication approach one step further and wrote a general form in terms of x (the number of hours) and then was able to solve part d algebraically. We discussed how the Kayak problem is powerful because it can be used in different grade levels to develop different skills. We talked about how all solutions are correct, but some solutions can be more efficient than others.

Beth discussed some norms for the classroom. We are here to learn to be fluent in mathematics, so everyone should expect to struggle a bit! It’s ok to make mistakes and learn from your mistakes, and it’s OK to feel challenged.

After the break, participants completed a pretest.

After lunch, we dove into the curriculum. We discussed the meaning of the equals sign, highlighting the importance of units (Compare “1 foot = 12 inches” to “1 = 12”) and taking care when using multiple equals signs in one row. We also started a discussion of fractions, highlighting how confusing the diagrams we use to introduce fractions can be in some cases. We saw how subdividing a line can be used to illustrate the idea of equivalent fractions.



  1. Complete page 1-16, “Rules that Expire” and “Language that Expires”.
  2. Read pages 1 – 11 from Principles to Actions.