Category Archives: Questions

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

I can establish mathematics goals to focus learning #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.

Establish mathematics goals to focus learning.

Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.

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 establish mathematics goals to focus learning, we refer to 5 Practices for Orchestrating Productive Mathematics Discussions by Peg Smith and Mary Kay Stein and Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning by John Hattie, Douglas Fisher, and Nancy Frey along with Principles to Actions: Ensuring Mathematical Success for All by Steve Leinwand.

To deepen our understanding around establishing mathematics goals, we anticipate, connect to prior knowledge, explain the mathematics goals to learners, and teach learners to use these goals to self-assess and level up.

From  NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions, we know that we should do the math ourselves, predict (anticipate) what students will produce, and brainstorm what will help students most when in productive struggle and when in destructive struggle.

Once prior knowledge is activated, students can make connections between their knowledge and the lesson’s learning intentions. (Hattie, 44 pag.)

To strengthen our understanding of using mathematics goals to focus learning, we make the learning goals visible to learners, ask assessing and advancing questions to empower students, and listen and respond to support learning and leveling up.

Excellent teachers think hard about when they will present the learning intention. They don’t just set the learning intentions early in the lesson and then forget about them. They refer to these intentions throughout instruction, keeping students focused on what it is they’re supposed to learn. (Hattie, 55-56 pag.)

How might we continue to deepen and strengthen our ability to advance learning for every learner?

What if we establish mathematics learning goals to focus learning?

Cross posted on Easing The Hurry Syndrome


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.

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.

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.

#KSULit2018: Mathematizing Read Alouds

At the 27th Annual KSU Conference on Literature for Children and Young Adults where the theme was Reimagining the Role of Children’s and Young Adult Literature, I presented the following 50-minute session on Tuesday, March 20, 2018.

Mathematizing Read Alouds

How might we deepen our understanding of numeracy using children’s literature? What if we mathematize our read-aloud books to use them in math as well as literacy? We invite you to notice and note, listen and learn, and learn by doing while we share ways to deepen understanding of numeracy and literacy.

Let’s debunk the myth that mathematicians do all work in their heads.  Mathematicians notice, wonder, note, identify patterns, ask questions, revise thinking, and share ideas.  Mathematicians show their thinking with details so that a reader understands without having to ask questions.

What if we pause during read-alouds to give learners a chance to analyze text features, to notice and wonder, to ask and answer questions in context?

How might we inspire and teach learners to make their thinking visible so that a reader understands?

Here’s my sketch note of the plan:

Here are more of the picture books highlighted in this session:

And, a list by approximate grade levels:

Early Learners, Pre-K, and Kindergarten

Kindergarten and 1st Grade

2nd, 3rd, 4th Grade

4th, 5th, 6th Grade

#T3IC: Using technology alongside #SlowMath to promote productive struggle

At the 2018 International T³ Conference in San Antonio, Jennifer Wilson (@jwilson828) and I presented the following two hour power session.

Using technology alongside #SlowMath
to promote productive struggle

How might we shift classroom culture so that productive struggle is part of the norm? What if this same culture defines and embraces mistakes as opportunities to learn? One of the Mathematics Teaching Practices from the National Council of Teachers of Mathematics’ (NCTM) “Principles to Actions” is to support productive struggle in learning mathematics. We want all learners to make sense of tasks and persevere in solving them. The tasks we select and facilitate must offer opportunities for each learner to develop connections and deepen their conceptual understanding.

Join us to learn more about #SlowMath opportunities that encourage students to persevere through challenging tasks instead of allowing their struggle become destructive. This session will address:

  • How might we provide #SlowMath opportunities for all students to notice and question?
  • How do activities that provide for visualization and conceptual development of mathematics help students think deeply about mathematical ideas and relationships?

Here’s the agenda:

8:30 Introductions
8:40 Intent and Purpose

  • Principles to Actions
  • #SlowMath
  • Norms (SMPs)
8:45 3-2-1 Bridge Visible Thinking Routine
8:50 Using Structure to Solve a Task – Circle-Square Task

9:55 3-2-1 Bridge Visible Thinking Routine

  • 2 questions around Productive Struggle (share one with partner and listen to one of partner)
10:00 Construct a Viable Argument to make your thinking visible:
Does (x+1)²=x²+1?

10:25 3-2-1 Bridge Visible Thinking Routine

  • In the chat, 1 analogy/metaphor/simile for Productive Struggle
10:30 Close

Here’s my sketch note of our plan:

Dave Johnston (@Johnston_MSMath) recorded his thinking and learning and shared it with us via Twitter.

And, a little more feedback from Twitter:

Cross posted on The Slow Math Movement

#SlowMath: looking for structure and noticing regularity in repeated reasoning #T3IC

At the 2018 International T³ Conference in San Antonio, Jennifer Wilson (@jwilson828) and I presented the following 90 minute
session.

#SlowMath: looking for structure
and noticing regularity in repeated reasoning

How do we provide opportunities for students to learn to use structure and repeated reasoning? What expressions, equations and diagrams require making what isn’t pictured visible? Let’s engage in tasks where making use of structure and repeated reasoning can provide an advantage and think about how to provide that same opportunity for students.

Here’s my sketch note of our plan:

Dave Johnston (@Johnston_MSMath) recorded his thinking and learning and shared it with us via Twitter.

Cross posted on The Slow Math Movement