Today we began by discussing the Coloring Fun! (yellow-red-blue patterns) “Burning the candle” problems.
After lunch we made a giant iceberg poster representing how ideas at each of our grade levels build floating capacity for understanding linear relationships such as those in the candle burning problem.
Then we spent some time exploring and sharing resources we found on the NCTM website with our new memberships!
Then we talked more about other kinds of relationships between quantities besides proportional: linear and non-linear alike!
Read the Professionalism Section of Principles to Actions (pp. 99- 108).
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?
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:
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
We are excited to announce the 2018 Summer Workshops.
- Math as a Second Language (Week 1). July 16-20, 2018. University Park.
- Algebraic Thinking and Proportional Reasoning (Week 2). July 23-27. University Park.
Participation in these workshops is free, but space is limited to 24 participants. Participants will receive 30 credit-hours per week toward continuing education (Act 48) requirements. Selected participants may also receive compensation for travel or lodging, based on need. Due to funding considerations, we do not expect to be able to provide stipends as in past years.
Selection is not first-come-first-served. Participants are chosen with the intent to balance grade bands, district need, geography, and other factors.
Applications are now open. See the 2018 Workshops page for more details and an application.