Category Archives: Algebra

Embolden Your Inner Mathematician: Week 2 agenda

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.
  Principles to Actions: Ensuring Mathematical Success for All

Slide deck

7:15 Establishing Intent, Purpose, Norm Setting

8:00 Continuing Talking Points – Elizabeth Statmore (@chessemonkeysf)

8:15 Number SplatsSteve Wyborney (@SteveWyborney)
8:25 Fraction SplatsSteve Wyborney (@SteveWyborney)
8:45 Planning for Splats

9:00 Closure and Reflection

  • I learned to pay attention to…
  • I learned to ask myself…
  • A new mathematical connection is…
9:15 End of session


  • Elicit and use evidence of student thinking using Splats. What will/did you learn?
  • Write to describe you quest for Closest to One using Open Middle worksheet with I can show my work so a reader understands without asking me questions.
  • Deeply Read pp. 207-211 from TAKING ACTION: Implementing Effective Mathematics Teaching Practices in K-Grade 5
    • What the Research says: Elicit and Use Evidence of Student Thinking
    • Promoting Equity by Eliciting and Using Evidence of Student Thinking
  • Read one of the following from TAKING ACTION: Implementing Effective Mathematics Teaching Practices in K-Grade 5
    • pp.183-188 Make a Ten
    • pp.189-195 The Odd and Even Task
    • pp. 198-207 The Pencil Task


Kelemanik, Grace, and Amy Lucent. “Starting the Year with Contemplate Then Calculate.” Fostering Math Practices.

Kaplinsky, Robert, and Peter Morris. “Closest to One.” Open Middle.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 46) Print.

Smith, Margaret Schwan., et al. Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5. The National Council of Teachers of Mathematics, 2017.

Statmore, Elizabeth. “Cheesemonkey Wonders.” #TMC14 GWWG: Talking Points Activity – Cultivating Exploratory Talk through a Growth Mindset Activity, 1 Jan. 1970.

Wyborney, Steve. “The Fraction Splat! Series.” Steve Wyborney’s Blog: I’m on a Learning Mission., 26 Mar. 2017.

Sketch notes from #TMC17 (a.k.a. Twitter Math Camp)

Becky Holden (@bholden86) and I attended Twitter Math Camp (#TMC17) at Holy Innocents Episcopal School in Atlanta, GA from Thursday, July 27 to Sunday, July 30.

This conference is by teachers, for teachers. The structure of TMC contains the following lengths of presentations:

  • Morning sessions (One session that meets Thursday, Friday and Saturday mornings for 2 hours each morning)
  • Afternoon sessions (Individual 1/2 hour sessions on Thursday)
  • Afternoon sessions (Individual 1 hour sessions Thursday, Friday and Saturday)

To honor Carl Oliver‘s (@carloliwitter) #PushSend request/challenge, here are my sketch notes from the sessions I attended.

Differentiating CCSS Algebra 1
— from drab to fab using Exeter Math 1 & Exploratory Talk
Elizabeth Statmore (@cheesemonkeysf)

The Politics(?) of Mathematics Teaching
Grace Chen (@graceachen)

What does it mean to say that mathematics teaching is political, and what does that mean for our moral and ethical responsibility as mathematics teachers?

Bridging elementary skills & concepts to high school & beyond
Glenn Waddell, Jr. (@gwaddellnvhs)

Micro-decisions in Questioning
David Petersen (@calcdave)

All I Really Need To Know I Learned From The MTBoS
…Not Really, But Close
Graham Fletcher (@gfletchy)

Hitting The Darn ‘Send’ Button
Carl Oliver (@carloliwitter)

Practical Ideas on the Kind of Coaching
We Need to Provide and Demand
Steve Leinwand (@steve_leinwand)

What is not captured in my notes is play: game night, trivia, crocheting, and tons of fun.

How might we grow, learn, and play in community when together and when apart?

Read, apply, learn

Read, apply, learn
`2017 T³™ International Conference
Saturday, March 11, 8:30 – 10 a.m.
Columbus H, East Tower, Ballroom Level
Jennifer Wilson
Jill Gough

How might we take action on current best practices and research in learning and assessment? What if we make sense of new ideas and learn how to apply them in our own practice? Let’s learn together; deepen our understanding of formative assessment; make our thinking visible; push ourselves to be more flexible; and more. We will explore some of the actions taken while tinkering with ideas from Tim Kanold, Dylan Wiliam, Jo Boaler and others, and we will discuss and share their impact on learning.

[Cross posted at Easing The Hurry Syndrome]

NCSM 2016: Sketch notes for learning

NCSM 2016 National Conference – BUILDING BRIDGES BETWEEN LEADERSHIP AND LEARNING MATHEMATICS:  Leveraging Education Innovation and Research to Inspire and Engage

Below are my notes from each session that I attended and a few of the lasting takeaways.

Day One

Keith Devlin‘s keynote was around gaming for learning. He highlighted the difference in doing math and learning math.  I continue to ponder worthy work to unlock potential.  How often do we expect learners to be able to write as soon as they learn? If we connect this to music, reading, and writing, we know that symbolic representations comes after thinking and understanding.  Hmm…Apr_11_NCSM-Devlin

The Illustrative Mathematics team challenged us to learn together: learn more about our students, learn more about our content, learn more about essentials for our grade and the grades around us.  How might we learn a lot together?

Conference Sketch Note - 25

Graham Fletcher teamed with Arjan Khalsa. While the title was Digital Tools and Three-Act Tasks: Marriage Made in the Cloud, the elegant pedagogy and intentional teacher moves modeled to connect 3-act tasks to Smith/Stein’s 5 Practices was masterful.
Conference Sketch Note - 26

Jennifer Wilson‘s #SlowMath movement calls for all to S..L..O..W d..o..w..n and savor the mathematics. Notice and note what changes and what stays the same; look for and express regularity in repeated reasoning; deepen understanding through and around productive struggle. Time is a variable; learning is the constant.  Embrace flexibility and design for learning.

Conference Sketch Note - 27

Bill McCallum challenges us to mix memory AND understanding.  He used John Masefield’s Sea Fever to highlight the need for both. Memorization is temporary; learners must make sense and understand to transfer to long-term memory.  How might we connect imagery and poetry of words to our discipline? What if we teach multiple representations as “same story, different verse”?

Conference Sketch Note - 28

Uri Treisman connects Carol Dweck’s mindsets work to nurturing students’ mathematical competence.  Learners persist more often when they have a positive view of their struggle. How might we bright spot learners’ work and help them deepen their sense of belonging in our classrooms and as mathematicians?

Conference Sketch Note - 29

Day Two

Jennifer Wilson shared James Popham’s stages of formative assessment in a school community. How might we learn and plan together? What if our team meetings focus on the instructional core, the relationships between learners, teachers, and the content?

Conference Sketch Note - 30

Michelle Rinehart asks about our intentional leadership moves.  How are we serving our learners and our colleagues as a growth advocate? Do we bright spot the work of others as we learn from them? What if we team together to target struggle, to promote productive struggle, and to persevere? Do we reflect on our leadership moves?

Conference Sketch Note - 31

Karim Ani asked how often we offered tasks that facilitate learning where math is used to understand the world.  How might we reflect on how often we use the world to learn about math and how often we use math to understand the world in which we live? Offer learners relevance.

Conference Sketch Note - 32

Day Three

Zac Champagne started off the final day of #NCSM16 with 10 lessons for teacher-learners informed from practice through research. How might we listen to learn what our learners already know? What if we blur assessment and instruction together to learn more about our learners and what they already know?

Conference Sketch Note - 33

Eli Luberoff and Kim Sadler created social chatter that matters using Desmos activities that offered learners the opportunities to ask and answer questions in pairs.  How might we leverage both synchronous and asynchronous communication to give learners voice and “hear” them?

Conference Sketch Note - 34

Fred Dillon and Melissa Boston facilitated a task to highlight NCTM’s Principles to Actions ToolKit to promote productive struggle.  This connecting, for me, to the instructional core.  How might we design intentional learning episodes that connect content, process and teacher moves? How might we persevere to promote productive struggle? We take away productive struggle opportunities for learners when we shorten our wait time and tell.

Conference Sketch Note - 35

Fluency: comprehension, accuracy, flexibility, and efficiency

No strategy is efficient for a student who does not yet understand it. (Humphreys & Parker, 27 pag.)

If both sense and meaning are present, the likelihood of the new information getting encoded into longterm memory is very high. (Sousa, 28 pag.)

When we teach for understanding we want comprehension, accuracy, fluency, and efficiency. If we are efficient but have no firm understanding or foundation, is learning – encoding into longterm memory – happening?

We don’t mean to imply that efficiency is not important. Together with accuracy and flexibility, efficiency is a hallmark of numerical fluency. (Humphreys & Parker, 28 pag.)

What if we make I can make sense of problems and persevere in solving them and I can demonstrate flexibility essential to learn?


Flexibility #LL2LU

If we go straight for efficiency in multiplication, how will our learners overcome following commonly known misconception?

common misconception: (a+b)²=a² +b²


correct understanding: (a+b)²=a² +2ab+b²

The strategies we teach, the numeracy that we are building, impacts future understanding.  We teach for understanding. We want comprehension, accuracy, fluency, and efficiency.

How might we learn to show what we know more than one way? What if we learn to understand using words, pictures, and numbers?

What if we design learning episodes for sense making and flexibility?

Humphreys, Cathy, and Ruth E. Parker. Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10. Portland, ME: Steinhouse Publishers, 2015. Print.

Sousa, David A. Brain-Friendly Assessments: What They Are and How to Use Them. West Palm Beach, FL: Learning Sciences, 2014. Print.

#LL2LU Mathematical Communication at an early age (TBT Remix)

Continue the pattern:  18, 27, 36, ___, ___, ___, ___

Lots of hands went up.

18, 27, 36, 45, 54, 63, 72
Yes! How did you find the numbers to continue the pattern?

S1:  I added 9.
(Me: That’s what I did.)
S2: I multiplied by 9.
(Me: Uh oh…)
S3: The ones go down by 1 and the tens go up by 1.
(Me: Wow, good connection.)

Arleen and Laura probed and pushed for deeper explanations.

S1: To get to the next number, you always add 9.
(Me: That’s what I did.)
S2: I see 2×9, 3×9, and 4×9, so then you’ll have 5×9, 6×9, 7×9, and 8×9.
(Me: Oh, I see! She is using multiples of 9, not multiplying by 9. Did she mean multiples not multiply?)
S3: It’s always the pattern with 9’s.
(Me: He showed the trick about multiplying by 9 with your hands.)

Without the probing and pushing for explanations, I would have thought some of the children did not understand.  This is where in-the-moment formative assessment can accelerate the speed of learning.

There were several more examples with probing for understanding. Awesome work by this team to push and practice. Arleen and Laura checked in with every child as they worked to coach every learner to success.  Awesome!

24, 30, 36, ___, ___, ___, ___
49, 42, 35, ___, ___, ___, ___
40, 32, 24,  ___, ___, ___, ___

I was so curious about the children’s thinking.  Look at the difference in their work and their communication.

By analyzing their work in the moment, we discovered that they were seeing the patterns, getting the answers, but struggled to explain their thinking.  It got me thinking…How often in math do we communicate to children that a right answer is enough? And the faster the better??? Yikes!  No, no, no! Show what you know, not just the final answer.

My turn to teach.

It is not enough to have the correct numbers in the answer.  It is important to have the correct numbers, but that is not was is most important.  It is critical to learn to describe your thinking to the reader.

How might we explain our thinking? How might we show our work? This is what your teachers are looking for.

The children gave GREAT answers!

We can write a sentence.
We can draw a picture.
We can show a number algorithm. (Seriously, a 4th grader gave this answer. WOW!)

But, telling me what I want to hear is very different than putting it in practice.

It makes me wonder… How can I communicate better to our learners? How can I show a path to successful math communication? What if our learners had a learning progression that offered the opportunity to level up in math communication?

What if it looked like this?

Level 4
I can show more than one way to find a solution to the problem.  I can choose appropriately from writing a complete sentence, drawing a picture, writing a number algorithm, or another creative way.

Level 3
I can find a solution to the problem and describe or illustrate how I arrived at the solution in a way that the reader does not have to talk with me in person to understand my path to the solution.

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.

Learning progression in 4th grade student-friendly language from Kato Nims (@katonims129)
Learning progression developed by #TrinityLearns 2nd Grade community learners-of-all-ages.

What if this became a norm? What if we used this or something similar to help our learners self-assess their mathematical written communication? If we emphasize math communication at this early age, will we ultimately have more confident and communicative math students in middle school and high school?

What if we lead learners to level up in communication of understanding? What if we take up the challenge to make thinking visible? … to show what we know more than one way? … to communicate where the reader doesn’t have to ask questions?

How might we impact the world, their future, our future?

#LL2LU Mathematical Communication at an early age was originally published on October 30, 2013.

SMP-6: attend to precision #LL2LU

Screen Shot 2015-04-09 at 4.19.47 PM

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

I can attend to precision.

But what if I can’t attend to precision yet? What if I need help? How might we make a pathway for success?

Level 4:
I can distinguish between necessary and sufficient conditions for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

 How many times have you seen a misused equals sign? Or mathematical statements that are fragments?

A student was writing the equation of a tangent line to linearize a curve at the point (2,-4).  He had written:  y+4=3(x-2)

And then he wrote:

Screen Shot 2015-01-02 at 5.33.27 PM

He absolutely knows what he means: y=-4+3(x-2).

But that’s not what he wrote.

Which student responses show attention to precision for the domain and range of y=(x-3)²+4? Are there others that you and your students would accept?

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How often do our students notice that we model attend to precision? How often to our students notice when we don’t model attend to precision?

Screen Shot 2015-02-15 at 7.07.43 PM

Attend to precision isn’t just about numerical precision. Attend to precision is also about the language that we use to communicate mathematically: the distance between a point and a line isn’t just “straight” – it’s the length of the segment that is perpendicular from the point to the line. (How many times have you told your Euclidean geometry students “all lines are straight”?)

But it’s also about learning to communicate mathematically together – and not just expecting students to read and record the correct vocabulary from a textbook.

[Cross posted on Easing the Hurry Syndrome]