Tag Archives: flexibility

#ShowYourWork, make sense and persevere, flexibility with @IllustrateMath

How might we learn to show our work so that a reader understanding without having to ask questions? As we work with our young learners, we want them to grow as mathematicians and as communicators.

We ask students to show their work so that a reader understands without having to ask them questions. What details should we add so that our thinking is visible to others?

To show (and to assess) comprehension, we are looking for mathematical flexibility.

I taught 6th grade math today while Kristi and her team attended ASCD.  She asked me to work with our students on showing their work.  Here’s the plan:

Learning goals:

  • I can use ratio and rate reasoning to solve real-world and mathematical problems.
  • I can show my work so that a reader can understanding without having to ask questions.


Learning progressions:

Level 4:
I can demonstrate mathematical flexibility with ratio and rate reasoning to show what I know more than one way using tables, equations, double number lines, etc..

Level 3:
I can use ratio and rate reasoning to solve real-world and mathematical problems.

Level 2:
I can make tables of equivalent ratios relating quantities with whole-number measurements, and I can use tables to compare ratios.

Level 1:
I can use guess and check to solve real-world and mathematical problems.

Anticipated solutions:

Sample student work:

Deep Dive into Standards of Mathematical Practice

As a team, we commit to make learning pathways visible. We are working on both horizontal and vertical alignment.  We seek to calibrate our practices with national standards.

On Friday afternoon, we met to take a deep dive into the Standards of Mathematical Practice. Jennifer Wilson joined us to coach, facilitate, and learn. We are grateful for her collaboration, inspiration, and guidance.

The pitch:

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The plan:


  • I can anticipate Standards for Mathematical Practice that learners will employ during this lesson.
  • I can begin to design lessons incorporating national standards, a learning progression, and a formative assessment plan.


  • Safe space
    • I can talk about what I know, and I can talk about what I don’t know.
    • I can be brave, vulnerable, kind, and considerate to myself and others while learning.
  • Celebrate opportunities to learn
    • I can learn from mistakes, and I can celebrate what I thought before and now know.


Learning Plan:

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The learning progressions:

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The slide deck:

As a community of learners, we

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Flexibility and adaptability: partial-quotients algorithm

How do we teach flexibility, adaptability, and multiple paths to success? How do we teach collaboration and taking the path of another?

Last Friday, I had an opportunity to teach my school’s 5th graders.  The lesson was on long division and the partial-quotients algorithm.  I took a 3-pronged approach. At the end of the lesson, I wanted these young learners to be able to say

    • I can take organized notes that will be useful when I study.
    • I can divide using more than one method.
    • I can show connections between the traditional algorithm and the partial-quotients algorithm.

I know that exceptional math students can show their understanding more than one way.  Most of these young learners prefer one method over the other.  I asked them to work in their bright spot strength first and then challenge themselves to work the same problem with the other algorithm.

I know that working with more than one method is a struggle for many learners.  Once I’ve found success, why would I try another method?  Sigh…why do some resist trying new things?

In Algebra I, we want learners to use the point-slope form of a line, the slope-intercept form of a line, and the standard form of a line.  We meet quite a bit of resistance to any method other than the slope-intercept form of a line.

What if we work on flexibility and adaptability ? How might we challenge ourselves to embed this expectation? It is not enough to find an answer. How many ways can I show what I know? What if I listen to a different point of view and try another path?