Category Archives: Learning

Traverse session: Experiential and Instructional: Promoting Productive Mathematical Struggle #tvrse18

At Traverse Boulder, I facilitated the following session on Tuesday, June 5, 2018.

Experiential and Instructional:
Promoting Productive Mathematical Struggle

How might we implement tasks that promote reasoning and problem solving to deepen conceptual understanding? Let’s identify and implement high quality tasks grounded in real experiences. Advancing the teaching and learning of mathematics cannot be accomplished with decontextualized worksheets. Discuss, sketch, and solve tasks that promote flexibility, creative and critical reasoning, and problem solving. Learning math should be anchored in depth of understanding through context – not pseudo context – and built on conceptual understanding as well as procedural fluency.

Here’s my sketch note of our plan:

Here’s the slide deck:

Just say no to worksheets.

Say YES to productive struggle and grappling.

Embolden your inner storyteller and leverage the art of questioning.

Context is key.

I can elicit and use evidence of student thinking #NCTMP2A #LL2LU

We strive to grow in our understanding of the Eight Mathematics Teaching Practices from NCTM’s Principles to Actions: Ensuring Mathematical Success for All. This research-informed framework of teaching and learning reflects a core set of high leverage practices and essential teaching skills necessary to promote deep learning of mathematics.

Elicit and use evidence of student thinking.

Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

In order to support our teaching teams as they stretch to learn more, we drafted the following learning progressions. We choose to provide a couple of pathways to focus teacher effort, understanding, and action.

When working with teacher teams to elicit and use evidence of student thinking, we refer to 5 Practices for Orchestrating Productive Mathematics Discussions by Peg Smith and Mary Kay Stein and Dylan Wiliam’s Embedding Formative Assessment: Practical Techniques for K-12 Classrooms along with Principles to Actions: Ensuring Mathematical Success for All by Steve Leinwand.

To deepen our understanding around eliciting evidence of student thinking, we anticipate multiple ways learners might approach a task, empower learners to make their thinking visible, celebrate mistakes as opportunities to learn, and ask for more than one voice to contribute.

From  NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions, we know that we should do the math ourselves, anticipate what learners will produce, and brainstorm how we might select, sequence, and connect learners’ ideas.

How will classroom culture grow as we focus on the five key strategies we studied in Embedding Formative Assessment: Practical Techniques for F-12 Classrooms by Dylan Wiliam and Siobhan Leahy?

  • Clarify, share, and understand learning intentions and success criteria
  • Engineer effective discussions, tasks, and activities that elicit evidence of learning
  • Provide feedback that moves learning forward
  • Activate students as learning resources for one another
  • Activate students as owners of their own learning

We call questions that are designed to be part of an instructional sequence hinge questions because the lessons hinge on this point. If the check for understanding shows that all students have understood the concept, you can move on. If it reveals little understanding, the teacher might review the concept with the whole class; if there are a variety of responses, you can use the diversity in the class to get students to compare their answers. The important point is that you do not know what to do until the evidence of the students’ achievement is elicited and interpreted; in other words, the lesson hinges on this point. (Wiliam, 88 pag.)

To strengthen our understanding of using evidence of student thinking, we plan our hinge questions in advance, predict how we might sequence and connect, adjust instruction based on what we learn – in the moment and in the next team meeting – to advance learning for every student. We share data within our team to plan how we might differentiate to meet the needs of all learners.

How might we team to strengthen and deepen our commitment to ensuring mathematical success for all?

What if we anticipate, monitor, select, sequence, and connect student thinking?

How might we elicit and use evidence of student thinking to advance learning for every learner?

Cross posted on Easing the Hurry Syndrome


Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 21) Print.

Stein, Mary Kay., and Margaret Smith. 5 Practices for Orchestrating Productive Mathematics Discussions. N.p.: n.p., n.d. Print.

Wiliam, Dylan; Leahy, Siobhan. Embedding Formative Assessment: Practical Techniques for F-12 Classrooms. (Kindle Locations 2191-2195). Learning Sciences International. Kindle Edition.

#NCTMLive #T3Learns Webinar: Establish Mathematics Goals to Focus Learning, and Elicit and Use Evidence of Student Thinking.

On Wednesday, March 28, 2018, Jennifer Wilson (@jwilson828) and I co-facilitated the first webinar in a four-part series on the Eight Mathematics Teaching Practices from NCTM’s Principles to Actions: Ensuring Mathematical Success for All.
        .
Establish Mathematics Goals to Focus Learning, and
Elicit and Use Evidence of Student Thinking.

Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

  • How might we communicate with clarity to ensure that learners are focused on high quality mathematical goals?
  • What types of tasks provide opportunities for learners to notice, note, wonder, and take action as agents of their own learning?
                 .
Our slide deck:
Agenda:
7:00 Opening remarks

  • Share your name and grade level(s) or course(s).
  • Norm setting and Purpose
7:05 Establish mathematics goals to focus learning #LL2LU

7:10 Let’s Do Some Math:  Illustrative Math – Fruit Salad?

7:25 Quotes from Taking Action
7:30 Elicit and use evidence of student thinking #LL2LU

7:35 Let’s Do Some Math

7:45 Elicit and use student thinking – Social-Emotional
Talking Points – Elizabeth Statmore

7:55 Close and preview next webinar in the series.

Implement tasks that promote reasoning and problem solving, and use and connect mathematical representations.

Some reflections from the chat window:
  • I learned to pay attention to how my students may first solve the problem or think about it prior to me teaching it to try and see connections that are made or how I can meet them. ~C Heikkila
  • I learned how to pay attention to how I introduce tasks to students. Sometimes I place limits on their responses by telling them what I expect to see in their responses as it relates to content topics. I will be more mindful about task introduction. ~M Roland
  • I learned to pay more attention to mathematical operations, and to look for more solutions that can satisfy the given problem. ~B Hakmi
  •  I also learned the importance of productive struggle and to be patient with my students. ~M James
  • I’m thinking about how to encourage my teachers to intentionally teach the mathematical practices. ~M Hite
  • I learned to pay attention to the learning progressions so I can think of the work as a process and journey. ~B Holden
  • A new mathematical connection for me was the idea of graphing values for the product example. ~A Warden
  • I learned to pay attention to peer discussions to discover how well students are learning the concepts. ~M Grech
  • Am I anticipating the roadblocks to learning? ~L Hendry

An audio recording of the webinar and the chat transcript can be viewed at NCTM’s Partnership Series page.

Cross posted at Easing the Hurry Syndrome

The self-discipline to wait, watch, coach (revised)

My small extended family, there are just 10 of us, blazed through 12 dozen homemade cookies in three afternoons. Home for the holiday, my mother, my daughter, and I bake for pleasure, to help the house smell good, and to pass on important family traditions.  The cookie baking extravaganza has now extended into day 4. The demand for more cookies might be triggered by the smell of chocolate, peanut butter, and sugar wafting throughout the house, back porch, and driveway. Or, it could be gluttony. It’s a holiday; calories don’t count, right?

In her new red and green pjs, AS wakes up raring to go.  Jumping up and down in the kitchen in her new polka dot apron, she asks “How many cookies will we bake today, Mama? How many? How many?”

I hold back a sigh and try not to drop my head; I am tired. I have turkey, dressing, ham, and several casseroles to prepare to carry on our traditions, and I am experiencing cookie overload. I muster my best smile and say, “We need to bake at least 4 dozen cookies. Uncle Jack is coming today, and you know how much he loves your cookies.”

It is our 4th day of cookie baking. Once again, by popular request, we were making Reese’s peanut butter cup cookies.  We make peanut butter cookie dough, roll it into balls, and cook them in mini muffin pans.  As they come out of the oven, we press mini Reese’s peanut butter cups into the center of the cookies.  Delicious.

It is day 4 of this algorithmic work.  The learner is still excited, curious, and engaged.  Am I? Do I feel the same engagement, or am I bored and ready to move on?

For the first 2 dozen, I make the batter, and three generations work together in concert to roll the cookies into balls. The tins come out of the oven holding peanut butter goodness just waiting to receive the Reese’s peanut butter cup candies.  Together, my mother, AS, and I press the candy into the cookies as they come out of the oven. I can still picture my grandmother’s hands doing this work with my mother and me.

Apprenticeship as learning is so important.

I am struck by the lessons my sweet 6-year old, AS, is teaching me about learning with my students. How often do our students watch us do the work to solve the problem or answer the question and pitch in at the last step?   

Baking the second 2 dozen is a very different story.  Thanks to my mother, AS her very own measuring spoons, spatula, and mini muffin pan that bakes 1 dozen muffins.  Empowered now that she has her own pan, she takes charge. It would have been so much faster for me to roll the cookies.  But, no…her pan; her cookies. Her mantra: “I can do it myself!”

So, I watch, wait, and coach.  I try not to cringe. I hold my comments so that I do not undermine her independence and confidence. Too small, the balls will be difficult to press candy into after baking in the oven.  Too big and they will blob out on the pan during baking. Patiently, I ask, “I wonder, honey, if the peanut butter cup will fit into that ball once baked. What do you think?” She fixes most of these problems with a little coaching from me.

Isn’t this happening in our classrooms?  It is so much faster and more efficient for the teacher to present the material.  We can get so much more done in the short amount of time we have. But, how much does the learner “get done” or learn?  When efficiency trumps learning, does anyone really have success? How do we encourage “I can do it myself!”? How do we find the self-discipline to watch, wait, and coach?

As she demands more independence, her confidence grows.  Can you believe that she would alter my recipe for the first 2 dozen cookies?  As our second dozen bakes, I press the peanut butter cups into my cookies. Miss I-Can-Do-It-Myself decides that Hershey kisses will be just as good or better.  With no prompting (or permission) she creates a new (to her) cookie. She has Hershey Kisses, and she wants to use them.

Worth repeating: “As she demands more independence, her confidence grows.” When we intervene too soon, are we stripping learners of their confidence and independence? Are we promoting productive struggle? Do we let them grapple enough?  

Does it really matter which method a learner uses to solve a problem or answer a question?  Isn’t it okay if they use the distributive property or an area model to multiply? Does it really matter which method is used to find the solution to a system of equations?  Shouldn’t they first find success? Don’t we want our learners to understand more than one way? Is our way always the best way?

Is AS pleased with herself and her creativity?  You bet. Are her cookies just as good as the original recipe?  Sure! How can you go wrong combining chocolate and peanut butter?

We must applaud the process that learners use to solve a problem or respond to a question.  We must praise them when they try something different. We must promote and encourage risk-taking, creativity, and problem-solving.

We must find the self-discipline to be patient while learning is in progress, to watch, wait, and coach.  We must embrace and promote the “I can do it myself!” attitude.

We must.


The self-discipline to wait, watch, coach was originally published on Dec 26, 2010.  This revision is inspired by what we are learning in Embolden Your Inner Writer.

I am grateful for the thoughtful, challenging, advancing feedback from Marsha Harris, Amanda Thomas, Kate Burton, Becky Holden, Cathrine Halliburton, and Lauren Kinnard.

#MVIFI Collider session Sketchnoting: Show what you know more than one way

At the February 16th MVIFI Collider event for professional learning, I facilitated the following 50-minute session on sketch noting twice.

Sketchnoting:
Show what you know more than one way

Up your note taking skills by being visual. Learn this invaluable method for recording, showcasing understanding, and deepening comprehension.

We will meet and greet, norm, touch on research, play with words and word art, discuss tools, practice, participate in a feedback look, and close by setting a micro-goal.

Here’s my sketch note of the plan:

We watched Simon Sinek’s TED talk to practice live sketch noting.

Here are artifacts of learning from Twitter:

Notice success, celebrate multiple milestones, level up

Learning intentions are more than just statements to convey to students what the learning is composed of; they are a means for building positive relationships with students. (Hattie, 48 pag.)

It is what I didn’t notice.  The bell rang. As always, I heard a chorus of “Thank you, Ms. Gough. Bye, Ms. Gough.” It was normal practice – and a much appreciated practice – for my students to say thank you and goodbye as they left for their next class.

I thought to myself “what a great class, everything went well, and they are so nice.” I busied myself straightening my desk, organizing paper, and mentally listing off the things I needed to do before my next class rolled in.  Eat lunch was at the top of the list.

Then, I sensed it. I was not alone.  It is what I didn’t notice.  There she sat, so still, except for the river of tears falling out of her beautiful, sad, green eyes. The river ran off the desk and pooled on the floor. “What is wrong?” I asked as I sat down beside her.

As I gently placed my hand on her arm, her shoulders began to shake as she said “I f..f..f..failed!” Whoosh, another flood of tears.

Now, she had not failed from my point of view. Her test score, damp as her test was now, showed a grade of 92 – an A.  And yet, she deeply felt a sense of failure.  As we sat together and looked at her work, we discovered that there was one key essential learning – in fact, a prerequisite skill – that caused her to stubble.

Tears, still streaming down her face, she said “I don’t know where I’m going wrong. I don’t miss this in class, but on the test, I fall apart.”

The point is to get learners ready to learn the new content by giving their brains something to which to connect their new skill or understanding. (Hattie, 44 pag.)

So, of course, the stumbling block for this sweet child is a known pain point for learners who master procedures without conceptual understanding.  Consistently, she expanded a squared binomial by “distributing” the exponent – a known pitfall. #petpeeve

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?

Kamb’s insight was that, in our lives, we tend to declare goals without intervening levels. We declare that we’re going to “learn to play the guitar.” We take a lesson or two, buy a cheap guitar, futz around with simple chords for a few weeks. Then life gets busy, and seven years later, we find the guitar in the attic and think, I should take up the guitar again. There are no levels. Kamb had always loved Irish music and had fantasized about learning to play the fiddle. So he co-opted gaming strategy and figured out a way to “level up” toward his goal:

Level 1: Commit to one violin lesson per week, and practice 15 minutes per day for six months.

Level 2: Relearn how to read sheet music and complete Celtic Fiddle Tunes by Craig Duncan.

Level 3: Learn to play “Concerning Hobbits” from The Fellowship of the Ring on the violin.

Level 4: Sit and play the fiddle for 30 minutes with other musicians.

Level 5: Learn to play “Promontory” from The Last of the Mohicans on the violin.

BOSS BATTLE: Sit and play the fiddle for 30 minutes in a pub in Ireland.

Isn’t that ingenious? He’s taken an ambiguous goal—learning to play the fiddle—and defined an appealing destination: playing in an Irish pub. Better yet, he invented five milestones en route to the destination, each worthy of celebration. Note that, as with a game, if he stopped the quest after Level 3, he’d still have several moments of pride to remember. (Heath, 163-164 pgs.)

What if I’d made my thinking visible?

What if I’d connected this learning to how 3rd graders are taught multiplication of two digit numbers by decomposing into tens and ones.  What if I’d connected this learning to how 3rd graders are also taught to draw area models to visualize the distributive property?

What if I’d shared my thinking and intentionally connected prior learning in levels?

By using Kamb’s level-up strategy, we multiply the number of motivating milestones we encounter en route to a goal. That’s a forward-looking strategy: We’re anticipating moments of pride ahead. But the opposite is also possible: to surface those milestones you’ve already met but might not have noticed. (Heath, 165 pag.)

How might we help our learners level up, experience success at several motivating milestones, and notice successes that might otherwise go unnoticed?

By multiplying milestones, we transform a long, amorphous race into one with many intermediate “finish lines.” As we push through each one, we experience a burst of pride as well as a jolt of energy to charge toward the next one. (Heath, 176 pag.)

Taken together, these practices make learning visible to students who understand they are under the guidance of a caring and knowledgeable teacher who is invested in their success. (Hattie, 48 pag.)


Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L.. Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Heath, Chip. The Power of Moments: Why Certain Experiences Have Extraordinary Impact. Simon & Schuster. Kindle Edition.

I’ll help you recover…

We will never know our reach unless we stretch. (Heath, 131 pag.)

When students don’t make errors, it’s probably because they already know the content and didn’t really need the lesson. (Hattie, 17 pag.)

Whack! One second everything was fine, then, for a fraction of a second, black. Kah-whop! I could see my phone, which used to be in my back pocket, hit the ice and slide about 8 feet in front of me.  Searing, hot pain surfaced in my left knee. It’s like I have a view from the ceiling. I can see myself face down on the ice. Cold. Wet.

I return to my eye’s view. I am really not sure what to do as I watch a nice soul skate over to my phone and bring it to me. While it was only a few seconds, it felt like 5 minutes of slow motion.  I was upright by then; no longer spread eagle face down on the ice.  

A sweet young thing glided up and laughed at me. “Ouch!” I heard myself say, “Don’t laugh! I’m hurt, and I don’t think I know how I’m gonna get up.” I saw her flinch but not leave me. My eyes confirmed that I was in a crowd and no one seemed to know what to do but stare.  #NotGood

The music teacher—“a woman with a beehive-ish hairdo and a seemingly permanent frown on her face”—led the choir in a familiar song, using a pointer to click the rhythm of the song on a music stand. Then, Sloop remembered, “She started walking over toward me. Listening, leaning in closer. Suddenly she stopped the song and addressed me directly: ‘You there. Your voice sounds . . . different . . . and it’s not blending in with the other girls at all. Just pretend to sing.’ ” The comment crushed her: “The rest of the class snickered, and I wished the floor would open and swallow me up.” For the rest of the year, whenever the choir sang, she mouthed the words. (Heath, 141 pag.)

Whatever momentary lapse in concentration caused me to fall – splat – did not feel good.  And the laugh, while meant to make light of an awkward situation, was crushing.  It was a mistake and a painful one at that.  

We hear it at school. We want our learners to be risk takers, to work on the edge of their ability, to fail faster, fail up, fail forward.  Right?

Get out there! Try something different! Turn over a new leaf! Take a risk! In general, this seems like sound advice, especially for people who feel stuck. But one note of caution: The advice often seems to carry a whispered promise of success. Take a risk and you’ll succeed! Take a risk and you’ll like the New You better!  That’s not quite right. A risk is a risk. (Heath, 131 pag.)

Errors help teachers understand students’ thinking and address it. Errors should be celebrated because they provide an opportunity for instruction, and thus learning. (Hattie, 16 pag.)

And just like that, she arrived.  An angel on the ice.  As she stretched out her hands, palms up, she said “Just take my hands.” I could get one foot square on the ice, though I felt like I was buried in a foot of snow, and then the other. Patiently she said “Now look at me and just press down.” I was up; shaken, but not broken.  Her beautiful brown eyes connected with mine and she smiled warmly as she said firmly “you are up and you are fine.” Just as quickly and elegantly as she arrived, she floated away.  

Then, in the summer after her seventh-grade year, she attended a camp for gifted kids in North Carolina called the Cullowhee Experience. She surprised herself by signing up to participate in chorus. During practice, she mouthed the words, but the teacher noticed what she was doing and asked Sloop to stick around after class. The teacher was short and thin, with hair down to her waist—a “lovely flower child,” said Sloop. She invited Sloop to sit next to her on the piano bench, and they began to sing together in the empty room. Sloop was hesitant at first but eventually lowered her guard. She said, “We sang scale after scale, song after song, harmonizing and improvising, until we were hoarse.” Then the teacher took Sloop’s face in her hands and looked her in the eyes and said: “You have a distinctive, expressive, and beautiful voice. You could have been the love child of Bob Dylan and Joan Baez.” As she left the room that day, she felt as if she’d shed a ton of weight. “I was on top of the world,” she said. Then she went to the library to find out who Joan Baez was. “For the rest of that magical summer,” Sloop said, she experienced a metamorphosis, “shedding my cocoon and emerging as a butterfly looking for light.” (Heath, 142 pag.)

My knee still throbbed and most of me was shaking.  I limped over to the edge of the rink until I could steady my nerves.  I’m not sure which hurt worse, my knee or my pride.  In either case, it hurt. But, I was up and I was fine.

The words, tones, facial expression, and body language we use with our learners matters.

Memorizing facts, passing tests, and moving on to the next grade level or course is not the true purpose of school, although sadly, many students think it is. School is a time to apprentice students into the act of becoming their own teachers. We want them to be self-directed, have the dispositions needed to formulate their own questions, and possess the tools to pursue them. (Hattie, 32 pag.)

How might we highlight what is going well for our young learners, accent the positive, and gently guide them to stretch, risk, and reach? What if we craft our feedback so our learners know we believe in their ability and expect great things even when they stumble, fall, and hurt? What if we guide their apprentice work to learn to use needed tools and hone their skills.

Our hopes and dreams for learning don’t include pretending – just stand there and mouth the words. Our learners must emerge as butterflies.

What type of feedback are we practicing? Laughter to make light of a stumble? Calm, “take my hand and push; you are fine?”

The promise of stretching is not success, it’s learning. (Heath, 131 pag.)

What great mentors do is add two more elements: direction and support. I have high expectations for you and I know you can meet them. So try this new challenge and if you fail, I’ll help you recover. (Heath, 123 pag.)


Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L.. Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Heath, Chip. The Power of Moments: Why Certain Experiences Have Extraordinary Impact. Simon & Schuster. Kindle Edition.