Category Archives: Daily Blog Recaps

Daily Recap: Week 1 Friday

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.

 

Daily Recap: Week 1 Thursday

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).

Blog Recap: Week 1 Monday (7/16/18)

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:

  • 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 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.

Math HW:

Refer to “Rules that Expire” (Block 1, p. 15 of notebook). For your assigned rules, follow the instructions on the sheet.

Pedagogy HW:

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.

PMI@UP 2017 Day 10

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.

Homework

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.

 

PMI@UP Summer 2017 Day 1

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).

  1. Conceptual Understanding – comprehension of mathematical concepts, operations, and relations.
  2. Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
  3. Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification.
  4. Strategic Competence –  ability to formulate, represent, and solve mathematical problems.
  5. 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.

Homework

Read Principles to Actions (the blue book):

  1. Sections titled Progress and Change; Effective Teaching & Learning (Pages 1-12)
  2. 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?

PMI@Berks Day 1 (6/26/17)

We are off to a great start for PMI at Berks.  We have 22 participants from 7 districts, with every grade level from Kindergarten to 6th represented.

This evening’s homework is as follows:

  • Read Principles to Actions pages 1-11 and pages 17-24.
    • Reflect on a task you have assigned your class, and analyze it from the perspective of what level of cognitive demand it makes.
  • Choose 3 categories from the “Spectrum of Subtraction” in Block 2, Page 9.  For each one, write a word problem appropriate for your grade level.  We will share these on Tuesday.
  • Complete both “Addition and Subtraction Problems” on Block 2, Page 10.

 

Day 5 of Functions & Algebra (7/29/16)

We started the morning with Andrea and Fran sharing resources. Andrea showed the Google Folder titled PMI 2016 Resources for Participants. Then she showed the NCTM website, and discussed benefits of membership (remember that K-8 institutions get a great deal!!). This segment ended with Andrea and Fran sharing some books (here are the covers):

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Andrew and Matt spent a brief time discussing Unit FA-4 on Functions, which are really just a formalization of the Processes from FA-1 with a new notation.  We then dipped into the Fox’s Furniture Store sequence from FA-5, both from the perspective of solving problems and evaluating students work on its mathematical merits.

Next we formed a large circle where Fran led a discussion about the readings on Professionalism from Principles to Actions.

After lunch, the group took the post-test (a necessary evil).  After that wrapped up Andrew and Matt tied up some loose ends and lingering questions: the Chipmunk formula from Wednesday’s homework, a strategy for solving the Milk Problem, and a justification for the “add the digits” trick for recognizing multiples of 9

It has been a fantastic two weeks!  We will be writing a longer post later with a full summary of these workshops and the workshops at Greater Allegheny and Brandywine as well.

Homework

Keep up your enthusiasm.  Seek change, and be patient.

Keep the commitment you made to yourself as you progress through the school year.

Don’t shy away from digging for the why of the mathematics you teach.  It all hangs together to form a cohesive whole.  You are welcome to send burning mathematical questions to Andrew.

Look for good candidates to recommend to PMI next summer.

Keep us updated on any revelations or experiences in your classroom that you can attribute to PMI.

Day 4 of Functions & Algebra (7/28/16)

Today we began with a discussion about the symbolic rules that describe the tortoise and hare problem. Fran emphasized that writing symbolic rules can be supported by working through the reasoning in other representations. Andrea shared some data about young children’s misunderstandings of the meaning of the equal sign and practices to avoid. We also watched a video of a young child using relational thinking to solve an open number problem.

We worked on the candle burning problem, which illustrates how a negative rate of change effects a linear relationship. After debriefing, Andrew talked through consequences of linearity.  We ended the morning with analyzing 12 situations to determine whether they could be solved via a linear relationship, a proportion, or neither.

After lunch we wrapped up some of our big pedagogy ideas by reflecting on last night’s messages, generating a list of things that need to happen in order to implement the strategies. We discussed more about the Standards for Mathematical Practice , focusing especially on supporting students to persevere.

We constructed a big iceberg wall together, representing ideas at each grade level that 20160729_104509contribute floating capacity to the concept of linear relationships.

 

We closed with a debriefing of the 12 situations on “What isn’t a linear relationship?” and highlighted the important features that distinguish them from each other.

See you tomorrow for our last day together!

Homework

Pedagogy: Read the section on Professionalism in Principles to Actions (pp. 99- 108).

Math:  None.

Day 3 of Functions & Algebra (7/27/16)

We are hip-deep in linear relationships (and a few things that aren’t).

Gallery Walk

We started the morning with posting some of the extension problems that participants wrote last night and then did a gallery walk.  Andrew highlighted two strategies for getting students to move to systematic or efficient strategies: using uglier numbers that are less prone to “I just knew it,” and multi-step problems where the task requiring the strategy is repeated multiple times and is part of a larger whole.

The Milk Problem

Then Andrea, Matt, and Andrew led a discussion of the milk problem, trying to make 2% milk from 1% milk and 4% milk.  A mathematical take-away lesson is “proportions don’t add, but amounts do.”  A meta-cognition lesson is “sometimes you just need to try something to see why it doesn’t work.”

Currency Conversion

Then Andrew continued the Currency Conversion problem, where now there is a fee involved.  We drew a lot of nice lessons out of it already:  parallel lines correspond to lines with the  same slope, the conversion rate corresponds to the slope of the line, the benefits of writing both fee-schemes as “convert to euros then subtract fee.”

Upside down teaching, real-world problems

We then moved into a pedagogy session, where we began with a discussion of the rich problems teachers developed during HW. Then each table generated a list for tomorrow’s iceberg activity (also from HW prompt). We also had a discussion of upside down teaching (from last night’s reading). Finally, we watched Dan Meyer’s TED talk.

Andrew shared the site mathpickle.com which has mathematically rich games and puzzles that students will happily engage in.

Terminology for Linear Relationships

Matt led a discussion putting names (slope, y-intercept, x-intercept) to concepts identified in the Currency Exchange with Fee problem.

The Tortoise and the Hare

We looked at the Tortoise and the Hare problem, wherein a scenario is presented and participants had to ask for certain information (e.g., the speeds) and not just handed it from the start.  Table-groups were then given different follow-up questions and asked to present their answers to the rest of the group.  Along the way we saw how many natural questions can be answered via a graph.

We finished the day be repeating the teaching episodes from Monday, but now practicing purposeful questioning and eliciting student thinking.

Homework for tonight:

Math

Write formulas for the Tortoise and the Hare problem that can answer this question: After x minutes, how many feet has the tortoise/hare/chipmunk run?  The following information was provided during class:

  • In the first minute the tortoise ran 1200 feet
  • In the first minute of running the hare ran 3000 feet.
  • The tortoise and hare maintain a constant speed.
  • The hare starts with a 10 minute nap, then runs.  The tortoise starts right away.
  • (Follow-up #7): A chipmunk starts from 12,000 feet behind the tortoise and runs at 2000 feet/minute.

Pedagogy

  1. Listen to the audio-recording of your “revised” teaching episode.
  2. In your notebook, write down the questions that you asked.
  3. Examine the questions you used while playing the role of teacher and write about what you learned about trying to use purposeful questions and eliciting student thinking.
  4. Read one of the messages from Smarter Than We Think. If your last name begins with A-M, read Message #14. If your last name begins with N-Z, read Message #16.
  5. Reflect back over all of the pedagogy discussions we have had in PMI and start a list in your notebook titled “Things I can do to help my students persevere when solving rich math problems.” Add at least three things to your list.

BTW: The “Messages” we’ve been reading come from the excellent book pictured below.

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