We began the day with some more work on linear and proportional relationships through the Purple Paint problems. Then we did Burning the Candle problems. We were especially interested in graphical representations of these scenarios and connected graphical representations to one another (red to blue; blue to red; total to red) and to other representations (tabular, symbolic, contextual). Then we discussed inverse relationships and we concluded by discussing at a “meta” level what things we might be interested in when describing “co-variation.”
After lunch we followed a jigsaw arrangement to discuss the 4 messages (readings are attached). These all come from Cathy Seeley’s messages books. Go here for more info about these books.
Then we spent time looking in our curricular materials, looking for ways we might start the year off well by raising the tasks’ level of cognitive demand, letting things go, incorporating review INTO new materials, supporting productive struggle, etc.
Then we classified sequence problems as linear, proportional, or neither, in order to better appreciate their attributes.
FIS ch. 16
FIS ch. 14
Let it Go..
Effectiveness and Efficiency
Today we talked about unit rates (and representations thereof) based work with the Coffee Maker problem and the Perfect Pint of Pink Paint. See the image below.
Then after lunch, we discussed the reading about “access and equity” from Principles to Actions. We discussed productive vs unproductive beliefs, we played with the scenario cards, and we connected all of that with
- the 5 strands of mathematical proficiency
- the Standards for Mathematical Practice
- The Effective Teaching Practices
Then we solved the Hourglass problem, and compared student work thereon.
Tonight you are invited to pre-read your assigned “message” reading. These were distributed at the end of class.
Have a good night and see you tomorrow!
This morning we spent learning about why the integer (positive and negative numbers) operations work! We talked about adding, subtracting, and multiplying signed numbers – ultimately learning why a negative number times a negative number is a positive number!!!
We then embarked on solving the Kayak problem. After lunch, we continued with the Kayak problem before moving on to pedagogy.
During the pedagogy time today, we read Cathy Seeley’s message titled “Upside Down Teaching” and talked about positive reactions to the message and challenges with teaching in this way. We then did the “Planning for Implementing High-Level Tasks” activity.
For those who want to reread, tomorrow we will be reading and discussing the Access and Equity section in PtoA (p. 59-69).
In the morning, we welcomed new teachers and started thinking algebraically! We revisted Otto and Hannah (Sibling Rivalry: Unknown Quantities), but this time they were eating Halloween candy, comparing heights, and running!
After lunch, we learned about performance and learning goals (Band Concert Task) and revisited how to use goals to support asking assessing/advancing questions (like we did last week with the Brownie Problem). We then did an activity where we analyzed a set of student work (Walking from School Task) and discussed how goals can support selecting, sequencing, and connecting student work in mathematical discussions (see page 30 of PtoA and the entire section on Facilitate Meaningful Mathematical Discourse). Fran ended the session with this diagram, which shows how goals can inform all of the other 7 PtoA teaching practices:
We ended the day with the “Back and Forth along Euclid Avenue” task – exploring positive and negative numbers! We then did Block 1, page 4 “Adding and Subtracting Signed Numbers via the Number Line.”
This morning we talked about multiplying fractions using the gardening plot problem. We discussed division, including division by zero.
Then we talked about talking about some “Rules that Expire,” but we didn’t actually talk about the rules themselves. 🙁
After lunch, we explored the NCTM website and resources. Then we discussed “Why can’t kids remember their basic facts?” and we read and discussed ways to help kids learn these facts. We talked about the developmental nature of learning facts (3 phases), the importance of using problem solving and strategies we can teach kids to help them reason.
We concluded with the modes of addition and subtraction and jogging and cutting cloth tasks for fraction division. (And how to “avoid” division).
We said good-bye to many new friends and colleagues, and we heard some words of wisdom from each of them.
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.