Today we
For homework, please
Read from the “Elementary and Middle School Mathematics” handout:
Today we
Talked A LOT about the dot problem
Generated a list of ideas about how to support productive struggle (using Ms. Ramirez and Ms. Flahive as examples):
Broke up into grade-level groups and began planning a “first-day lesson” by anticipating student responses
For tomorrow:
Read the Message called “Upside-down teaching”
In your notebook, complete respond to the discussion prompts for teachers at the end of the message (on p. 94). Try to make connections about what we’ve read about and discussed so far in PMI.
Many thanks to Kimberly for baking some delicious chocolate zucchini bread. Her recipe is below. (Bonus question: If you only want to make a half a loaf, how much shredded zucchini do you need? Write the number sentence to describe that scenario.)
Here’s what we did today
We also watched the video about “The uncomfortable effort of thinking.”
Here’s what’s due for tomorrow
Read Principles to Actions, the section titled “Support Productive Struggle in Learning Mathematics” (pp. 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?
Here’s what we did today:
Handshake problem: Solved it, watched 3rd graders work on it
Discussed addition and subtraction and the spectrum of scenarios that are addition and subtraction problems.
Discussed Levels of Cognitive Demand. (Addition strings and watching a 1st grade classroom)
We began our discussion of adjective-noun theme.
Here’s what’s due for tomorrow:
Read Principles to Actions: Section titled “Build Procedural Fluency from Conceptual Understanding” (pp. 42-48).
Also read: Webb, Bozwinkel, & Decker. Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding. MTMS, 2008. (Handout from class today)
In your notebook: Identify a procedure or skill that you consider essential for students at your grade level to learn. List the conceptual understandings that support students’ learning of the procedure or skill.
Write questions you have about the readings.
Write three problems illustrating different categories of addition and subtraction, as outlined from Block 2, page 8 “you try.”
Work through “Two Gross Problems” in Block 4, page 6.
Here are the definitions of the Strands of Mathematical Proficiency that were in the PowerPoint today:
Adding it Up: Helping Children Learn Mathematics, National Research Council (2001, p. 116).
This book can be downloaded for free at this link.
Homework
Read Principles to Actions (the blue book):
Write answers to the following prompts in your journal:
Read through and reflect on the 8 Rules that Expire.
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 
contribute 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!
Homework
Pedagogy: Read the section on Professionalism in Principles to Actions (pp. 99- 108).
Math: None.
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:
Pedagogy Homework:
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:
We got a group picture, which will be posted soon.
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.
We next moved into fraction multiplication by exploring 4 interpretations of (2/3)×(4/5):
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.
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!
We began the pedagogy part of the day by compiling suggestions for a “I’m Stuck” poster:
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.)
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.
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?
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)
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.