We are happy to announce we will be offering our usual 2-week workshop Summer 2019.
Dates: July 15-19 and 22-26. (9am – 4pm daily)
Location: Penn State University Park Campus, McAllister Building.
See the information page for more details and an application.
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.
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
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.
Update 7/23/18: We are sorry to announce that we will need to cancel the workshop announced below due to a lack of participants. We hope to offer a similar workshop at PSU Brandywine in the fall or winter.
We are happy to announce that we are offering a 3-day workshop at Penn State Brandywine in Media PA. Participation is free, and lunch is provided, but there are only 25 seats. Participants will be chosen on a first-come-first-served basis. Participants providing their PPID can earn 18 continuing education credits. Workshops will run July 31 – August 2, 8:30am-3:30pm, with lunch provided each day.
The workshops will focus on math content from grades 5-8, and will be open to both newcomers to PMI and to those who have already participated in prior PMI workshops. Topics to discuss include:
Making sense of decimals, negative numbers, rational numbers, and irrational numbers
Using ratios, proportions, and percentages flexibly to solve problems
Mathematical modeling with linear equations
Teaching through rich problems and discussion
Leveraging conceptual understanding to build procedural fluency
Unpacking the Standards of Mathematical Practice