Category Archives: Algebra

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?

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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.
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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.

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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”?

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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?

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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?

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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?

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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.

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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?

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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?

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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.

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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?

SMP-1-MakeSensePersevere

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

multiplication_flexibility

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.

show-your-work
Learning progression in 4th grade student-friendly language from Kato Nims (@katonims129)
IMG_8439
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

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

I can attend to precision.
(CCSS.MATH.PRACTICE.MP6)

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:

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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?

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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]

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)

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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?

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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?

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Or over the set of complex numbers? 

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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? 

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

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What if we are looking at powers of i?

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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]

 

 

How do we use the December exam as formative assessment? (TBT Remix)

“For assessment to function formatively, the results have to be used to adjust teaching and learning; thus a significant aspect of any program will be the ways in which teachers make these adjustments.”
Inside the Black Box: Raising Standards Through Classroom Assessment Black and Wiliam

How do students reflect on their work?  What opportunities are offered to help students carry the essential learnings from first semester through second semester and/or into the next level of learning?

I’m interested and curious about different strategies and methods used to help learners process and reflect on their exam experience and the accumulation of what they know.  Since each learner will have different bright spots and strengths, what strategies are used to differentiate for intervention and enrichment?

We aim to get “in the weeds” about reflection and intervention.  We want every child to reflect on what they could demonstrate well and where they need additional help.  We do not want them to move to the next year with any doubt or weakness if we can help now.  But, how do we know who needs help?  We collect data, and we let our learners gather data.  We need to be informed; they need to be informed.  We are a team working toward the goal of mastery or proficiency for all learners.

Our process:

  1. Return the exam to the learner on the first day back.
  2. Have each learner complete the exam analysis and reflection form (shown below) to identify strengths and areas of need.
        1. Circle the number of any missed problem.
        2. Begin, and possibly complete, correcting missed problems to review the material and determine if any error was a simple mistake or if more help is needed.
        3. Write a reflection about strengths, struggles, and goals.
        4. Report results on our team’s Google doc. (This is a copy; feel free to explore and “report” data to see how it feels. You can view the results here.)
  3. Meet in team to review all results and analyze for groups to design and provide necessary intervention and additional learning experiences.
  4. All assessments 2nd semester will have questions from first semester essential learnings to offer learners the opportunity to show growth and to help with retention.

We would love it if others would share methods and strategies for helping learners grow from an exam experience.  How do students reflect on their work?

What opportunities are offered to help students carry the essential learnings from first semester through second semester and/or into the next level of learning?


How do we use the December exam as formative assessment? was originally posted on January 4, 2012.