Category Archives: Learning

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.

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?

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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.

 

Productive struggle and essential feedback

How might we teach and learn more about perseverance? I wish we could rephrase the first Standard for Mathematical Practice to the following:

I can make sense of tasks and persevere in solving them.

Just the simple exchange from problems to tasks make this process standard a little more global for learners.  What if we encourage and expect productive struggle?

Some struggle in learning is good, but there is a key distinction to be made between productive struggle and destructive struggle.  Productive struggle allows students the space to grapple with information and come up with the solution for themselves. It develops resilience and persistence and helps students refine their own strategies for learning. In productive struggle, there is a light at the end of the tunnel; learning goals not only are clear but also seem achievable. Although students face difficulty, they grasp the point of the obstacles they face and believe that they will overcome these obstacles in the end.(Jackson and Lambert, 53 pag.)

How might we make a slight change during the learning process to challenge our learners, to promote productive struggle, to persevere, and to learn, through experience, critical reasoning?

But many people are petrified of bad ideas. Ideas that make us look stupid or waste time or money or create some sort of backlash. The problem is that you can’t have good ideas unless you’re willing to generate a lot of bad ones.  Painters, musicians, entrepreneurs, writers, chiropractors, accountants–we all fail far more than we succeed. (Godin, n. pag.)

What if we reframe “failure” as productive struggle and perseverance?

  • Level 4:
    I can find a second or third solution and describe how the pathways to these solutions relate.
  • Level 3:
    I can make sense of tasks and persevere in solving them.
  • Level 2:
    I can ask questions to clarify the problem, and I can keep working when things aren’t going well and try again.
  • Level 1:
    I can show at least one attempt to investigate or solve the task.

We cannot emphasize enough the power of feedback. Given the right kind of feedback, struggling students can gauge how they are doing and determine what they need to do to get to mastery. It can help students quickly correct their mistakes, select a more effective learning strategy, and experience success before frustration sets in. (Jackson and Lambert, 68 pag.)

How might we highlight many paths to success? What if we make paths to success visible enough for learners to try, risk, question, and learn?

When people believe their basic qualities can be developed, failures may still hurt, but failures don’t define them.   And if abilities can be expanded – if change and growth are possible – then there are still many paths to success.” (Dweck, 39 pag.)


Dweck, Carol S. Mindset: the New Psychology of Success. New York: Random House, 2006. 39. Print.

Godin, Seth. “Seth’s Blog: Fear of Bad Ideas.” Seth’s Blog. N.p., n.d. Web. 06 Feb. 2015.

Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series). Association for Supervision & Curriculum Development. Kindle Edition.

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.

Fractions with unlike denominators – a lesson plan

Vicki graciously allowed me to teach our 5th graders again today.  Kerry and Marsha gave their time to observe and offer me feedback.

Learning Progression 

Level 4:
I can show what I know in numbers and pictures.

Level 3:  
I can use equivalent fractions to add and subtract fractions.

Level 2:
I can use visual models to add and subtract fractions.

Level 1:
I can decompose fractions into the sum or difference of two fractions.

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We used the Navigator to collect responses from all learners prior to going over each task.

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Think of the questions and the peer-to-peer discussions.  There are as many students answering 4/8 as 14/15.  Can you describe the thinking that might yield an answer of 4/8?

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How might we work toward making thinking visible? Why is peer-to-peer discourse so important? What if we practice flexibility to show what we know more than one way?

#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.

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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.

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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?”

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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?

Teachnology: Using technology “differently” (TBT Remix)

When we ponder how, when and why to integrate technology, do we consider how learners might use digital tools as instruments of self-assessment, feedback, and tinkering to learn?

Last week I was “schooled” in using technology by a first grader.

agough

She was invited to write for edu180atl.  Her post was published on 5.2.12.  To draft her post, we selected two pictures to use as inspiration.  She wrote a story for each picture and selected one for submission.  HOW she used technology to write was a HUGE lesson for me.

She took my computer from me and wrote 3 sentences.  There was a word that had a red “crinkly” line under it.

 

The instant feedback transitioned the technology to teachnology; it caused her to ask herself questions.  Finally, she asked me how to spell inspired.  Then, she read her 3 sentences out loud and decided that she needed another sentence in between two of the current sentences.  (Do I do that when I write?)

She was determined to have 200 words, not 198 words or 205 words.  She wanted 200 words exactly.  She learned how to use the word count feature since both stories were in the same document.  She read out loud and deleted words.  She read out loud again and added words.  It was awesome to watch.  She chose to ask to have a “peer” editor.  “Are there 2 words that I can delete? I want exactly 200 words.”  How much more confidence would I have about my writing if I had published articles and ideas when I was younger?

This experience with my first grader makes me wonder about learning – well, anything – with technology.  What assumptions do we make about what learners will and won’t learn if we put technology in their hands?

“How can we focus on what we do best without missing new opportunities to do better?” (Davidson, 17 pag.)

_________________________

Davidson, Cathy N. “I’ll Count-You Take Care of the Gorilla.” Introduction. Now You See It: How the Brain Science of Attention Will Transform the Way We Live, Work, and Learn. New York: Viking, 2011. 17. Print.


LEARNing: Using technology “differently” was originally posted on May 14, 2012