Category Archives: Questions

Differentiation and mathematical flexibility – #LL2LU

How is flexibility encouraged and practiced? Is it expected? Is it anticipated?  What if we collect evidence of mastery of flexibility along side mastery of skill?

From Jo Boaler’s How to Learn Math: for Students:

…we know that what separates high achievers from low achievers is not that high achievers know more math, it is that they interact with numbers flexibly and low achievers don’t.

This past week Rhonda Mitchell (@rgmteach), Early Elementary Division Head, and I collaborated to reword the learning progression for mathematical flexibility so that it is appropriate for Kindergarten and 1st Grade learners.

How might we differentiate to deepen learning?

If we want to support students in learning, and we believe that learning is a product of thinking, then we need to be clear about what we are trying to support. (Ritchhart, Church, and Morrison, 5 pag.)

How might we collect evidence to inform and guide next steps?

Monitoring students’ mastery of a learning progression leads to evidence collection for each building block in a progression. (Popham, Kindle location 2673)

How might we prepare for mid-course corrections to intervene, enrich, and personalize learning for every learner?

By learning to insert feedback loops into our thought, questioning, and decision-making process, we increase the chance of staying on our desired path. Or, if the path needs to be modified, our midcourse corrections become less dramatic and disruptive. (Lichtman, 49 pag.)

What if we consider pairing a skill learning progression with a process learning progression? How might we differentiate to deepen learning?

Students love to give their different strategies and are usually completely engaged and fascinated by the different methods that emerge. Students learn mental math, they have opportunities to memorize math facts and they also develop conceptual understanding of numbers and of the arithmetic properties that are critical to success in algebra and beyond. (Boaler and Williams)

Boaler, Jo, and Cathy Williams. “Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts.” Youcubed at Stanford University. Stanford University, 14 Jan. 2015. Web. 22 Feb. 2015.

Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

Popham, W. James (2011-03-07). Transformative Assessment in Action: An Inside Look at Applying the Process. Association for Supervision & Curriculum Development. Kindle Edition.

Ritchhart, Ron, Mark Church, and Karin Morrison. Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners. San Francisco, CA: Jossey-Bass, 2011. Print.

#TEDTalkTuesday: Math in an Information Age

Conrad Wolfram: Teaching kids real math with computers
We can engage so many more students with this, and they can have a better time doing it. And let’s understand: this is not an incremental sort of change. We’re trying to cross the chasm here between school math and the real-world math. 
Dan Meyer: Math class needs a makeover
 Math makes sense of the world. Math is the vocabulary for your own intuition.

Learning progressions: formative assessment, learning tactics

The leaner a learning progression is, the more likely it is to be used. (Popham, Kindle Location 485)

Screen Shot 2015-02-06 at 7.16.14 PM

What if we use learning progressions to empower students and personalize learning?

It’s the teacher who establishes the expectation and the conditions so that each student can monitor his or her own learning progress and decide whether or not to make a learning tactic adjustment. (Popham, Kindle Locations 291-292)

The benefits of it, however, can be remarkable—namely, students who are actively reviewing their own classroom assessment data, connecting these outcomes to their own inputs, and making changes so that their efforts will yield more satisfactory results. (Popham, Kindle Locations 294-295)

Screen Shot 2015-01-30 at 3.40.31 PM Screen Shot 2015-01-30 at 3.45.26 PM

Popham, W. James (2011-03-07). Transformative Assessment in Action: An Inside Look at Applying the Process. Association for Supervision & Curriculum Development. Kindle Edition.

Moving to productive struggle

From “Mrs. Maas, how do I do this?” to “I finished and helped a friend.

How might we engage more learners simultaneously, offer visible opportunities to show what they know, and personalize feedback, intervention, and enrichment?

Screen Shot 2015-02-06 at 7.16.14 PM

What if we offer learners pathways to guide progress, actions, and collaboration?  What if we encourage productive struggle by offering guidance about process, actions, and collaboration? What if we intervene with coaching?

In case you cannot read Becky‘s learning progression above, I’ve included an edited version of it here:

  • Level 4:
    I can complete my item, and I can help others with theirs, explaining the circuit.
  • Level 3:
    I can build a wired item for Mom with materials provided.
  • Level 2:
    I can plan a wired item (layout and switch) with help from classmates or Mrs. Maas.
  • Level 1:
    I can get ideas from others on a plan.

Becky guides learners to plan, collaborate, test their independence, and then, when possible, contribute to the success of others. And, through the process, learn about circuits too.

“Collaboration by difference respects and rewards different forms and levels of expertise, perspective, culture, age, ability, and insight, treating difference not as a deficit but as a point of distinction.”  (Davidson, 100 pag.)

When our learners do not know what to do, how do we respond? What actions can we take – will we take – to deepen learning, empower learners, and to make learning personal?

Davidson, Cathy N.  Now You See It: How the Brain Science of Attention Will Transform the Way We Live, Work, and Learn. New York: Viking, 2011. Print.


Maybe we need to think of it as teachnology rather than technology. (TBT Remix)

The time with our learners is limited.  We have to make some very important decisions about how to use this time.  We must consider the economics of our decisions based on the resources we have.  Is it cost effective, cognitively, to spend multiple days on a learning target to master something that a machine will do for us?

Is what we label as problem-solving and critical thinking really problem-solving and critical thinking or is it just harder stuff to deal with?  Can we teach problem-solving and critical thinking in the absence of context?

Do we have a common understanding of what good problem solvers and critical thinkers look like, sound like, and think like?  If we are teaching problem-solving, critical thinking, and creativity, shouldn’t we know what that means to us?  Shouldn’t we be able to describe it?

Does technology hamper or enhance a learner’s ability to problem solve and think critically?  I think I might be back to the struggle of using calculators to compute and a spell checker to write.  Do we even know enough to make a decision about technology until we experiment and learn by doing?

If you have not read Can Texting Help Teens with Writing and Spelling? by Bill Ferriter, stop reading this right now to read Bill’s post.  It is a great example of leveraging technology to promote creativity and critical thinking using technology.  Read about having students write 25 word stories.  This is teachnology, not technology.  Tweet, text, type, write on paper – it doesn’t matter – unless you want to publish your work.  The technology, Twitter in this case, aids in the critical thinking; you are restricted to 140 characters.  The technology offers the learner a way to publish and see other published work.

My ability to transport myself from place to place is actually enhanced and improved because of my truck.  I have no idea how my truck works other than gas goes in, step on the brake to stop, R means we are going to go in reverse, etc.  I do not need to understand the mechanics; I can have that done.

I do not need to understand the mechanics; I can have that done.   I don’t need to know how to change the oil in my car.  I need to know that I need to have the oil changed in my car.  And, very important, I don’t need to learn this lesson by experience.  It is too expensive to learn experientially why I must have the oil changed in my car.

Isn’t it too expensive to spend 2-3 days on some topics that we traditionally teach?  Are we getting the biggest bang for our cognitive buck?  Often our learners can’t see the forest for the trees.  They never get to the why because of the how.  Don’t we need to learn when and how to use technology not only to engage our learners, but to increase our cognitive capital?

How can we learn to ask

  • Why are we learning this?  Is this essential?
  • Will technology do this for us so that we can learn more, deeper?
  • Does this have endurance, leverage, and relevance?
  • Shouldn’t we use technology to grapple with the mechanics so the learner shifts focus to the application, the why, the meaning?

Maybe we need to think of it as teachnology rather than technology was originally posted on January 26, 2011.

#TEDTalkTuesday: Making ideas clear by visualizing them

Tom Wujec: 3 ways the brain creates meaning

We make meaning by seeing, by an act of visual interrogation. The lessons for us are three-fold. First, use images to clarify what we’re trying to communicate. Secondly make those images interactive so that we engage much more fully. And the third is to augment memory by creating a visual persistence. These are techniques that can be used to be – that can be applied in a wide range of problem solving.

So the act of collectively and collaboratively building the image transforms the collaboration.

Sunni Brown: Doodlers, unite!

There are four ways that learners intake information so that they can make decisions. They are visual, auditory, reading and writing and kinesthetic. Now in order for us to really chew on information and do something with it, we have to engage at least two of those modalities, or we have to engage one of those modalities coupled with an emotional experience. The incredible contribution of the doodle is that it engages all four learning modalities simultaneously with the possibility of an emotional experience. 

Steve Johnson: Where good ideas come from

#LL2LU Show your work – grade 4

How might we foster a community of learners where everyone bravely and fiercely seeks feedback?

I was at EduCon in Philadelphia when this tweet arrived last week.

Screen Shot 2015-01-30 at 3.40.31 PM

Am I showing enough work? How do I know? What if we partner, students and teachers, to seek feedback, clarity, and guidance?Screen Shot 2015-01-30 at 3.45.26 PM

Success inspires success.

Yesterday, I dropped by Kato‘s classroom to work on the next math assessment and found our learners working together to apply math and to improve communication.

Screen Shot 2015-01-30 at 3.54.14 PM

Now, I was just sneaking in to drop off and pick up papers.  But, how could I turn down requests for feedback?

Here’s the #showyourwork #LL2LU progression in the classroom:

Grade 4

Level 4
I can show more than one way to find a solution to the problem.

Level 3
I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2
I can find a correct solution to the problem.

Level 1
I can ask questions to help me work toward a solution to the problem.

And here’s one child and her work. “Ms. Gough, will you look at my work? Can you understand it without asking me questions? Is is clear to you?”

Screen Shot 2015-01-30 at 4.10.55 PM

I see connected words, pictures, and numbers. I like the color coding for the different size bags.  I appreciate reading the sentences that explain the numbers and her thinking.  I also witnessed this young learner improve her work and her thinking while watching me read her work.  She knew what she wanted to add, because she wished I knew why she made the final choice.  I’d call this Level 4 work.

What if we foster a community of learners who bravely and fiercely seek feedback?