Category Archives: Learning

Visual: SMP-8: look for and express regularity in repeated reasoning #LL2LU

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.
(CCSS.MATH.PRACTICE.MP8)

Screen Shot 2015-04-04 at 5.15.38 PM

But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

[Cross posted on Easing the Hurry Syndrome]


Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.

SMP-8: look for and express regularity in repeated reasoning #LL2LU

Screen Shot 2015-04-04 at 5.03.13 PM

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning. (CCSS.MATH.PRACTICE.MP8)

But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

What do you notice? What changes? What stays the same?

Can we use CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning?

What do we need to factor for the result to be (x-4)(x+4)?
What do we need to factor for the result to be (x-9)(x+9)?
What will the result be if we factor x²-121?
What will the result be if we factor x²-a2?

Screen Shot 2015-04-05 at 10.10.10 AM

We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1. (And then connect the result to the graph of y=x²+1.)

What happens if we factor over the set of real numbers?

Screen Shot 2015-04-05 at 10.12.39 AM

Or over the set of complex numbers? 

Screen Shot 2015-04-05 at 2.17.14 PM

What about expanding the square of a binomial? 

What changes? What stays the same? What will the result be if we expand (x+5)²?  Or (x+a)²?  Or (x-a)²? 

Screen Shot 2015-04-05 at 2.17.56 PM

What about expanding the cube of a binomial?  Or expanding (x+1)^n, or (x+y)^n?

Screen Shot 2015-04-05 at 2.18.14 PM

What if we are looking at powers of i?

Screen Shot 2015-04-05 at 2.18.31 PM

We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Easing the Hurry Syndrome]

 

 

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?

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.

 

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.

This slideshow requires JavaScript.

photo 2

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

Screen Shot 2015-02-03 at 8.24.34 PM

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?

IMG_1274

Screen Shot 2015-02-03 at 12.02.34 PM

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?