We started the day with a long discussion of adding and subtracting fractions and mixed numbers with different denominators. This was built on the same principles of renaming that serve us well when working with differing units and place value.
We then looked over the mathematics homework from the night before on the Purple Punch problem and shared the variety of solutions at our tables. We then worked through a 3-act math task (Nana’s chocolate milk) and then started on the sequence of problems on currency conversion.
After lunch, Andrea and Fran led a session grounded in the brownie problem (7 brownies shared with 4 people). We developed learning and performance goals for the task. Then we read about assessing and advancing questions, examined student work for the brownie problem, and developed assessing and advancing questions for those students.
We then got is small groups and shared about the readings last night (messages 14 and 16) about “Effectiveness and Efficiency” and “Letting it Go.” We had a good whole group talk about these two messages.
We ended the day with some light mathematical coloring according to a described pattern. The goal is to create a procedure by which we can quickly determine the color any given number will receive.
Homework: No pedagogy homework for Thursday. Math homework is to continue to analyze and record your observations about the coloring problem.
Today was our field trip to Chambers Building!
This morning, we began our discussion of ratios in earnest. We began with the coffee problem, then moved into the “Perfect Pint of Pink Paint” problem. Through these we could see how ratios can be represented in multiple ways, as well as how ratios interact with multiplying by a little more and adding a little more.
Before lunch we welcomed Dean David Monk and Associate Dean Greg Kelly from the College of Education.
After lunch we discussed the patterns in our own questioning that we heard in our teaching recordings. We watched a video clip of an 8th-grade teacher working on the “Two Tanks” problem and analyzed her questioning.
Here is the list of question stems Fran promised to share.
For a caricature of funneling questions, see this short: https://www.youtube.com/watch?v=KdxEAt91D7k
Andrew showed another way to think about multiplication with integers, and we ended the day doing more proportion problems (“the perfect pitcher of purple punch”).
Read whichever short message was handed out to you in class today. (Either “Let it go” or “Effectiveness and efficiency”).
Revisit the “Perfect pitcher of purple punch” A-E in light of Laura’s bar model and a table-based approach.
This morning we began by discussing algebraic reasoning and reasoning with patterns. We used the Sibling Rivalry problem to illustrate how we can reason around an unknown baseline.
We then discussed modeling positive and negative numbers using a number line, addition and subtraction on that number line, and then we used colored chips to do the same. Thanks, Derrick!
After lunch, we discussed timed tests and teaching basic facts. We read together from this selection from Van de Walle, Karp, and Bay-Williams. Andrea recommended two articles from Teaching Children Mathematics: one about assessing basic facts by Kling and Bay-Williams, and one about the importance of thinking by Buchholz.
From Principles to Actions (the blue book): Read two sections:
- Pose purposeful questions (p. 35)
- Elicit and use evidence of student thinking (p. 53)
In your notebook, write a response to these three prompts:
- 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.
- 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.
- 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?
We began by continuing our discussion of division with fractions. We drew pictures and looked for patterns in an effort to understand why it works to “invert and multiply.”
Then we reflected on the reading about “Upside-Down Teaching,” and brainstormed ways to shift from I-We-You approaches to You-We-I approaches.
We had time to plan for this shift by looking at curricular materials and plans for the first few weeks of school. We watched a video of the “My favorite no” practice.
We ate a delicious lunch together and some of us said good-bye to Fran. Sadface.
After lunch, Melina shared recommendations for the OGAP frameworks. Thanks, Melina!
We resumed work on operations with fractions. This time we pushed ourselves to develop even deeper understandings of the “invert and multiply” procedure.
We began by revisiting the “brownie” homework from Tuesday night, which called for us to critique student responses in light of “the whole.” This made us sensitive to the issue accounting for “the whole” as we worked through the Garden Plot and Land-Tax Task to illuminate “the rules” for multiplying fractional quantities. In the end we came to the following conclusions (among others):
- The array and area models introduced in grade 3 directly impact how one can understand fraction multiplication in grade 4.
- The numerator functions to count the “number of bits” and the denominator indicates size of the “bits” in terms of “how many bits to make 1 unit” (which may or may not be the total number of bits shown). Unlike the brownie problem, “the whole” is not arbitrarily chosen but rather is a consequence of “the unit” for each of the factors.
After lunch, we first created a list with this sentence stem: To support productive struggle, teachers can…
Then we discussed “learning goals” and “performance goals” (see handout from today). After that, we worked in grade-alike pairs/groups to plan for the implementation of a high-level task and supporting productive struggle.
Then we watched a video about mindsets and the impact of different kinds of praise. And Fran shared Jo Boaler’s book on mathematical mindsets:
Melina suggested that we check Amazon for grade-level books by Jo Boaler. Here is the link to the 3rd-grade book.
Following the discussion on productive struggle, we struggled productive ourselves as we grappled with the “Cut from the same cloth” problem, division, and how natural division of fractions is. At the tail end I mentioned the other post with good sources of rich problems.
Pedagogy HW: Read “Upside Down Teaching” by Cathy Seeley (handout).
- Think about how this message connects to the teaching practices that we have talked about this week from Principles to Actions: Implement tasks that promote reasoning and problem solving; Use and connect mathematical representations; Build procedural fluency from conceptual understanding; and Support productive struggle in learning mathematics. Make some notes to support your participation in group discussion tomorrow.
Math HW: Reflect on Division of Fractions and why the “keep-change-flip” trick is accurate (but short-cuts the conceptual underpinning).
The following is a (partial) list of rich problems
Illustrative Mathematics: https://www.illustrativemathematics.org/
(many rich tasks, keyed to CCSS codes)
Graham Fletchy’s website: https://gfletchy.com/
(includes 3-act tasks for elementary grades, progression videos, and other interesting resources)
(more to come, hopefully…)
Another fantastic day!
We began with the name game, with a total of 210 names said. We then shared our solutions to the Caterpillar problem, listing 4 different formulas and ideas for more.
We then moved into the “Gross Problem” to complicate our instincts on how place value interacts with subtraction. Solutions presented included subtracting mixed numbers and how that also draws upon similar ideas when borrowing.
Just before lunch we discussed using conceptual understanding to build procedural fluency.
After lunch we discussed the “iceberg” model, and how any topic is supported by many earlier ideas.
We then moved into multiplicative algorithms, in particular how the Partial Products (or area method) is tucked into both the standard algorithm and the lattice method.
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?
Watch the following videos which relate to productive struggle
Science of Thinking: https://www.youtube.com/watch?v=UBVV8pch1dM
Don’t Get Stuck: https://www.youtube.com/watch?v=VrSUe_m19FY
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:
- 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.
- 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:
A great first day!
We started with introductions, along with welcome messages from Mark Levi and George Andrews.
We then jumped into our first problem, the Star Spangled Banner Problem. Participants shared a variety of solution methods. These led to the following formulas:
Formulas for the number of stars after n days
We briefly touched on connections between these, and will return to that later.
We spent the rest of the morning on the pre-test and then participants got a chance to share their backgrounds and start networking before lunch.
After lunch, we discussed “Smarter than We Think” (by Cathy Seeley). We developed posters that listed “inspirations” and “challenges.”
Fran presented three frameworks that guide PMI’s design and implementation:
- Standards for Mathematical Practices: What kinds of activities should students engage in while learning mathematics? These are listed in Principles to Actions and defined on in your notebook in the front matter section on page 5.
- Eight Teaching Practices: How should teachers teach so that students engage in learning mathematics so that they become mathematical proficient? This is the main content of Principles to Actions: Ensuring Mathematical Success for All. We will unpack several of these teaching practices over the two weeks!
We ended the day with a discussion of the “Adjective Noun Theme,” which is based on the notion that numbers are most useful when they are considered with context. Numbers are adjectives describe the amount of some noun. We then tried to apply this viewpoint to compare similar fractions, treating the denominator as a noun and the numerator as an adjective.
Refer to “Rules that Expire” (Block 1, p. 15 of notebook). For your assigned rules, follow the instructions on the sheet.
Read Principles to Actions (the blue book): Sections titled “Progress and Change” and “Effective Teaching & Learning”(Pages 1-12)
- What did you learn from reading the “Progress and Change” section about the state of mathematics education in the U.S.? Write a couple of thoughts in your notebook.
- Think about: How did you react to the chart of beliefs on page 11? In your notebook, write a reflective response (a few sentences) to the beliefs chart.
This post is just to say that this is the place where we will be posting summaries of each day’s topics and the homework for the next day.