We began our morning by drying off and (in some cases) taking some drastic measures to dry our shoes.

Then we reviewed the homework revisiting the Kayak, Film, and Currency problems. We considered how the concepts of x-intercept, y-intercept, slope, and inverse function relates to each of those problem contexts. Josh got us started by showing and explaining his work on the Kayak problem. (Thanks Josh!)

We then moved into the discussion of the “What isn’t a linear relation?” sheet.  We came up with some key features that linear relations have (e.g., the constant rate of increase or decrease).  We also discussed proportions, and how to tell whether a linear relation is a proportion (in particular, the two quantities maintain the same ratio throughout the table).  Andrew admitted to providing a “rule that expired” in checking only whether the two quantities were zero at the same time (this broke down in the octagon problem).  We went down several side roads, including continuous vs discrete quantities, additive vs multiplicative comparisons, and how proportions are fundamentally multiplicative relationships.

Before lunch, Andrea and Fran recommended more books and resources:

1. Seeley, C. (2014). Smarter than we think: More messages about math, teaching, and learning in the 21st century. Sausalito, CA: Math Solutions.
2. Shumway, J. (2011). Number sense routines: Building numerical literacy every day in Grades K-3. Portland, ME: Stenhouse.
3. Teaching Children Mathematics- a monthly magazine for early childhood and elementary mathematics teachers (Go here for more info.)
4. Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006). Teaching number in the classroom with 4-8 year-olds. London: Paul Chapman.
5. Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston: Allyn & Bacon.
6. NCTM’s Navigating through…. .. (Part of the Principles and Standards for School Mathematics Navigations Series.) Go here for more info.

After lunch, we used Andrew’s adaptation of the Candle Burning problem as an opportunity to discuss ideas related to tasks and cognitive demands, funneling vs. focusing, scaffolding, productive struggle, and curriculum materials. We moved into a discussion of equity and access by discussing last night’s reading from Principles to Actions and the graphic of kids standing on boxes. We concluded the pedagogy discussion by posting our grade-level tasks in a timeline on the wall.

Then we worked on the “Fox’s Furniture Problem.”  Andrew started by presenting the scenario without a given question, eliciting participants to ask questions that struck them.  Participants worked on solving the problem and explore other possibilities.  We found commutativity and associativity provide a way to justify why a “30%, then 10%” discount yields the same final price as a “10%, then 30%” discount.  Terri showed how using a variable can demonstrate that the 40% discount will always be better than “10% then 30%” (or the other way), regardless of the original price.

We ended by analyzing some students’ work to solve the problem, and judged them based on accuracy and efficiency.  We found that a correct answer does not always imply a correct process, but also that a “weird” process is not necessarily inaccurate.  We discussed how re-trying the purported method for another set of numbers can often reveal that the process is wrong, but also that trying just one more example might lead to a weird coincidence (as happened in class).

Pedagogy Homework

Read the “Professionalism” section from Principles to Actions (pages 99-108).

Math Homework

Review your materials from the past two weeks in preparation for the course evaluation (also the post-test, I guess).

Also, at lunchtime, Andrew insisted that Fran and Andrea pose for this picture with him and the Lion:

Enjoy your last evening in State College!

Fran began the morning by sharing a number of resources:

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

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?

Today we began by “jigsawing” the homework problems from last night. Teachers were assigned to “expert groups” on one of the problems: partitive vs. quotitive division, changing perimeters of a rectangle with fixed area, and the handshake problem. Experts formed new groups to discuss the problems further. Andrew showed us how the graph of y = 24/x (a hyperbola) can help thinking about changing the dimensions for a fixed area. We watched a video of 3rd graders working on the handshake problem (mini Gausses!) We also heard some great ideas from colleagues about effective ways to organize students into groups and to provide extra work for groups of students who finish early and need differentiated learning opportunities.

We discussed many models to understand fractions from a variety of perspectives, not just the “pizza model.”    In particular we found that many of us overlook using fractions as distances or locations on the number line, but these other models can be of great use when justifying fraction arithmetic.  We used the area model to expand our ideas about equivalent fractions and fraction multiplication.  We also drew on the adjective-noun theme to justify the process of finding a common denominator.

We sat in a circle and practiced one another’s names.  We discussed productive classroom norms regarding engaging in mathematical problem solving, participating as productive group members, and what strategies teachers can use to help establish such norms.

After a lunch and picture-taking break, we resumed work with fractions.  We discussed why division of fractions should involve multiplying by the reciprocal.  In short we did it by the same way we came to understand division yesterday: translating a division problem with an unknown result into a multiplication with an unknown factor.

We ended by working on the sheet about estimating fractions, and developing fraction number sense.

We said good-bye to our colleagues who won’t be joining us next week! We’ll miss you, and we hope to see you next summer if not before!

Math and Pedagogy Homework:

No HW, but please bring as many of your curriculum materials as you can on Monday. We’ll be re-examining your textbooks together.

But…..if you’re looking for some math to do: How many names were said when we played the name game? That is: by the time we had made it around the whole circle and each of us had taken a turn, how many names had been uttered? Hmmm…….

Have a great weekend!

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: http://www.adjectivenounmath.com/.  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: https://www.teachingchannel.org/videos/class-warm-up-routine

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.

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:  https://www.youtube.com/watch?v=NWv1VdDeoRY

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: https://www.youtube.com/watch?v=hiiEeMN7vbQ

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?

Welcome to the 2015 PMI Workshops!

Today we…

We begin with a welcome from the group: Andrew Baxter, Fran Arbaugh, Andrea McCloskey, Charles Helou, and Beth Lindsey.  We also had a welcome from Math Department Chair Yuxi Zheng.  We then dove into the Kayaking on the Susquehanna River problem, and discussed (1) the issue of ambiguity in a written problem, (2) the multiple modes of solution, (3) how those solutions can relate to each other, and (4) how different methods have different advantages and disadvantages.  For example, the algebraic method was fast but without extra work it did not answer the more practical question of how to decide between venues for other lengths of time.  After the break, we spent the rest of the morning on the pre-assessment.

After lunch we discussed the “Smarter Than We Think” article. Then we read the beliefs chart on page 11 of Principles to Action and had small group discussions about our own beliefs about teaching and learning mathematics.  We will continue to discuss the ideas of a “growth mindset” and “fixed mindset” as the weeks go on.

We finished Unit 1, discussing what equality should mean, and how students misunderstand it.  We also began Unit 2 by discussing the adjective-noun theme and how it applies to addition.   We then applied the adjective-noun theme to see how it makes sense of the place-value system in addition and adding fractions with a common denominator.

Here are the definitions of the Strands of Mathematical Proficiency that were in the PowerPoint today:

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: Helping Children Learn Mathematics, National Research Council (2001, p. 116).
This book can be downloaded for free at:
http://www.nap.edu/catalog/9822/adding-it-up-helping-children-learn-mathematics

Handouts

• Paperwork to be completed.
• Front matter for Math as a Second Language (e.g., title page, table of contents, expectations)
• Unit 1: Fundamentals
• Unit 2: Perspectives on Addition

Homework

For Tuesday:

• Solve the Film Developing Problem(s)
  • Focus on problems 1 and 2.
  • Attempt at least two of the three variations in problems 3, 4, and 5, as time permits.
  • (Leave space in your notebook for discussion notes.)
• Optional: The riddles from Unit 1, and Guess My Noun from Unit 2.
• Read in Principles to Actions:
  • Sections titled “Progress and Change” and “Effective Teaching & Learning” (pages 1-12 – stop reading at the middle of page 12)
  • Section titled “Implement Tasks that Promote Reasoning and Problem Solving” (pages 17-24).
• Respond (in your notebook) to the following prompts:
  • Reflect on a typical in-class math lesson (use your textbook to refresh your memory) that you have taught. 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?
• Bring your math textbooks to PMI on Tuesday.