What a full two weeks! And yet there always seems to be more to do.
We started the morning with the post-assessment and beliefs survey. The analysis of that data will be shared later this summer.
We then spent some time refreshing the addition and subtraction of signed numbers, and representing that with joining and taking away piles of color-coded counters. That discussion progressed into exploring the multiplication of signed numbers. We found that when one or two of the factors are positive whole numbers, then we can rely on the “__ groups of size ___” model. However, when both factors are negative we need to rely on logic and the distributive property in particular. We ultimately had three explanations: one based on patterns, one based on the notion of the opposite, and one based on comparing the products 4×4 and the expanded form of (5+(-1))x(5+(-1)).
We then split into teams of 3 to do “Bungee Barbie” and started analyzing the data over lunch. Here is the winning drop: https://photos.app.goo.gl/TJO0BwptyT0slWrm2 . After the drops, we debriefed on the mathematical thinking and topics that occurred within that task.
We then got into our “Kumbaya” circle and reflected on what we learned about mathematics, what we learned about teaching, and our commitments to how we can improve our own classrooms. We then read “Hooray for Diffendoofer Day.” The key take-away message: Teach your students to think and the tests will take care of themselves.
Continue to give your students the best educational experiences possible. Continue to learn how to make those experiences richer, and continue to seek out why of mathematics.
Productively struggled our way through the candle burning problems as we learned about linear relationships
Watched some videos:
- Mindset #1: https://www.youtube.com/watch?v=NWv1VdDeoRY
- Mindset #2: https://www.youtube.com/watch?v=hiiEeMN7vbQ
- Escalator: https://www.youtube.com/watch?v=VrSUe_m19FY
- My favorite NO: https://www.teachingchannel.org/videos/class-warm-up-routine
Worked in grade level groups to made a big iceberg poster about linear relationships. How do the concepts we teach at each grade level build floating capacity for engaging with the candle burning problem?
Discussed parents and families, using the message from Cathy Seeley’s book.
Discussed negative and positive numbers.
For tomorrow, please:
Read the Professionalism section of Principles to Actions (pp. 99- 108)
We started the day with meeting the returning participants. We then moved to reviewing the homework at our tables. We proceeded to discuss some student work on the Hourglass Problem, and then the Punch Problem and attending to precision.
In the afternoon we discussed the importance of using rich problems (and keeping them rich when they meet the classroom). We then joined up with the returning participants and found out how they implemented what they’ve learned at PMI.
Make an attempt at the “Two Routes” problem, including the Going Further section.
No pedagogy homework.
This morning we dug deep into the question “How can we tell a pattern will always hold?” by revisiting the Snail problem, and then exploring the “Even and Odd” activity in Block 1.
In the afternoon we discussed the teaching practices of Pose Purposeful Questions and Elicit and Use Evidence of Student Thinking.
After the pedagogy discussion we began Block 2 and relating to proportional reasoning.
- Read either Message 14 or Message 16, based on your last name.
- Block 2, Page 3. Problems 1 and 4. Try to use as many representations of proportional relationships as possible.
- Moved onto the algebraic thinking units
- Discussed basic facts, drills, and timed tests (see the messages we generated, below). Here is the website Andrea shared that provides opportunities for selected practice opportunities for fluency development
- Discussed calculators in the math classroom
- Interviewed a classmate for tonight’s homework
Here are some websites with calculator lessons and activities:
- The Math Tools website (http://mathforum.org/mathtools) allows you to search by grade level and by the type of technology you wish to use.
- Texas Instruments has a collection of calculator activities to review (http://education.ti.com/ calculators/downloads/US/Activities/).
- Casio has several calculator activities to review for elementary and middle school (http://edu.casio.com/support/activity/).
Here are our messages to ourselves about basic facts:
- More work on strategies and practicing the strategies BEFORE drilling
- When a 9th grader is struggling, provide tools for remediation (teach strategies!). It’s not enough to say “learn them!”
- Timed tests are required by my district—use it as a learning tool (and assessment). Assess for individual fact families and find out what strategies they do/don’t use
- Some facts are more powerful than others
- Timed tests frustrate the struggling students even more and reward the quick thinkers
- “you’re competing against yourself, not each other”—emphasize personal growth (chart for themselves)
BTW: Here is a nice website about number strings.
For tomorrow, please
Read Principles to Actions. “Pose Purposeful Questions” (p. 35-41) and “Elicit and Use Evidence of Student Thinking” (p. 53-57)
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?
Try the Maze Playing Board. Let’s see who has the largest value tomorrow. There MAY be a prize involved.
- Drew pictures and solved problems related to fraction multiplication and division
- Discussed strategies for learning basic facts (addition and multiplication)
- Continued yesterday’s planning work in grade-level teams
For homework, please
Read from the “Elementary and Middle School Mathematics” handout:
- pp. 158-159 (Drill of Efficient…”), and
- pp. 174-end (Effective drill…”)
Talked A LOT about the dot problem
Generated a list of ideas about how to support productive struggle (using Ms. Ramirez and Ms. Flahive as examples):
- Have your room set up for collaboration
- Set the climate for day 1
- Celebrate mistakes (my favorite mistake)
- Use “can you catch my mistake” problems – Identify my “not yet” – analyze my answer – oops
- Be honest with kids – tell them what you are doing and why.
- Acknowledge that learning math/thinking is hard, but we can do it.
- Practice makes progress, not perfect
- Build stamina; start where they are; “Think” stage – start with a few seconds and then build up.
- Ask students to make a plan before starting “solving”
- Decorate your room with people who succeeded after “failing” several times.
- Have “hip pocket” responses “what are you thinking?”
- Figure out where kids might have struggles with the task.
- Stop talking so much.
- Make sure you have manipulatives available
- Have anchor charts
Broke up into grade-level groups and began planning a “first-day lesson” by anticipating student responses
Read the Message called “Upside-down teaching”
In your notebook, complete respond to the discussion prompts for teachers at the end of the message (on p. 94). Try to make connections about what we’ve read about and discussed so far in PMI.
Many thanks to Kimberly for baking some delicious chocolate zucchini bread. Her recipe is below. (Bonus question: If you only want to make a half a loaf, how much shredded zucchini do you need? Write the number sentence to describe that scenario.)
Here’s what we did today
- Discussed last night’s “gross problems” (and made connections to kids’ struggles with place value concepts)
- Examined a variety of strategies for solving addition and subtraction with multidigit numbers (“alternative algorithms”)
- Watched Dan Meyer’s TED talk: Math Class Needs a Makeover and discussed some of the “yeah, buts…” that emerged for us
- Made “Iceberg posters” for procedures at each of our grade bands (see the picture below)
- Generated a list of reasons to put conceptual development before procedural development:
- helps with retention
- helps with flexible use of #s
- rushing to fluency can cause anxiety and “bad” feelings about math
- builds on students’ prior knowledge
- conceptual dev. serves as a check for computationshttp://veritasium.com/education/the-uncomfortable-effort-of-thinking/
- in the real-world, problems are more likely to be more conceptual than procedural
We also watched the video about “The uncomfortable effort of thinking.”
Here’s what’s due for tomorrow
Read Principles to Actions, the section titled “Support Productive Struggle in Learning Mathematics” (pp. 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?
Here’s what we did today:
Handshake problem: Solved it, watched 3rd graders work on it
Discussed addition and subtraction and the spectrum of scenarios that are addition and subtraction problems.
Discussed Levels of Cognitive Demand. (Addition strings and watching a 1st grade classroom)
We began our discussion of adjective-noun theme.
Here’s what’s due for tomorrow:
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 class today)
In your notebook: Identify a procedure or skill that you consider essential for students at your grade level to learn. List the conceptual understandings that support students’ learning of the procedure or skill.
Write questions you have about the readings.
Write three problems illustrating different categories of addition and subtraction, as outlined from Block 2, page 8 “you try.”
Work through “Two Gross Problems” in Block 4, page 6.
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 this link.
Read Principles to Actions (the blue book):
- Sections titled Progress and Change; Effective Teaching & Learning (Pages 1-12)
- 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?
- Consider the chart about beliefs on page 11. What is your reaction to this description of unproductive and productive beliefs?
Read through and reflect on the 8 Rules that Expire.
- When does the rule work?
- Bonus: What grade level does that scenario occur?
- Give a mathematical scenario where the rule does not work (at least, not in the way it’s phrased)
- Bonus: What grade level does that scenario occur?