This morning we:
- Discussed the “sum of odds” problem (1 + 3 + 5 + …….+ 99)
- Built on that to work on the “Insatiable Caterpillar” problem
- Then we worked on the Piggy Bank Problems
- Then we played (and nailed!) the name game
After lunch we:
Discussed how developing conceptual understanding can support procedural fluency. We generated the list below:
- ”Concrete to abstract” : concrete action supports conceptual understanding, and this provides access to be able to model themselves
- Hands-on activities –> mental representations –> supports abilities to do procedures
- If only procedural skills: kids either remember or they don’t, and then don’t have resources
- “transfers” to other problems, other representations. THIS IS MATHEMATICAL THINKING
- Our job is to create problem-solvers. We have so many tools, kids need to know how and when and why to use them.
- Provides meaning (beyond ”just for the test”)
Watched video of Rachel’s renaming of a mixed number into an “improper fraction”
Then we developed iceberg posters representing concepts that build “floating capacity” for a variety of procedures.
We ended the day by naming the properties of multiplication. This related to today’s mathematical themes: There are many multiplication algorithms, each with their own advantages, and The methods of decomposing and recomposing are especially key to making sense of multiplication algorithms.
Read Principles to Actions section on productive struggle (pp 48-52).
Read the article we distributed called Supporting productive struggle with communication moves.
Spend 20 minutes on the “Handshake problem” on the Evening Reflection page of Day 3.
Our math for the morning explored the theme that “the same quantities have many names, and context determines the most useful name.” We focused on discussions of the equal sign, place-value, and fractions and algorithms for computing operations of addition and subtraction.
In the afternoon, Fran presented three frameworks that guide PMI’s design and implementation:
- Strands of Mathematical Proficiency: What does it mean to know mathematics? 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).
- 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.This book can be downloaded for free at: http://www.nap.edu/catalog/9822/adding-it-up-helping-children-learn-mathematics
- 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 also read the Principles to Actions section about “implementing tasks that promote reasoning and problem solving.” We discussed the levels of cognitive demand, and we generated a group list of strategies for tweaking low level tasks to raise their level of cognitive demand:
- Putting into real-life situation
- Use “justify” and “explain” in problem
- Open up multiple ways to do the problem
- Giving the answer and asking how you could “find” this answer (more than one
- Asking “how” questions instead of just “what” questions
- Use manipulatives or hands-on
- Use compare and contrast
- Make it less about “answer getting” and more about “math thinking”
- Use questions where there is not just one right answers (allow multiple correct
- Asking students to find “one more” way or answer
During our math for the afternoon, we explored combining collections of even and odd numbers.
On the Day 2 Evening Reflections, do everything but items #1 and #3. You also don’t need to do the “aliases of a fraction” prompt. [note: by “Reference” pages, Andrew means pages 4 and 5 from the packet.]
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 today)
Think about how these two readings connect and make some notes in your notebook.
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.
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?
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).