Evidence of (Professional) Learning

As I work to design and deliver professional learning for my educators, I continue to seek ways to improve their experiences.  I am pursuing ways for my educators to capture their learning while they are in the sessions, share this learning beyond the session, and access later once the session is over.  Two examples from this summer are the use of Clips + FlipGrid and Pages with placeholder images.

Clips + FlipGrid

One day-long session related to supporting Gifted and Talented students in mathematics included 50+ elementary and secondary educators.  Early in the session I requested the attendees download the Clips app if they had not done so already.  Throughout our time together I reminded the educators to take photos and videos of the process.  These artifacts were stored in their camera roll, ready to access near the close of the session.

As the session came to a close, the educators used Clips to summarize and reflect on their time in the session.   In 20-30 minutes, all 50+ educators successfully created summaries of their learning with photos and videos as evidence!  These videos were shared within and beyond our time together with FlipGrid.

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Pages with Placeholder Images

Two half-day sessions related to teaching students supported through Special Education services in mathematics included elementary and secondary educators as well.  A portion of the experience involved clarification of our standards.  The educators were challenged to collaboratively model a given strand of the standards using manipulatives, images, and text.  These artifacts were captured and organized on a Pages document.  I created the Pages document with placeholder images ahead of time and shared through iCloud.

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The educators used their camera to capture photos and successfully demonstrated understanding of the standards as they created this artifact of their learning!

Next steps of my work include supporting my teachers to collect evidence of learning as part of their bigger professional learning journey.

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NCSM 2018

Some of the most insightful, inspiring minds in math education converged upon Washington DC this week for the annual meeting of the National Council of Supervisors of Mathematics.  I left recharged and ready to continue to do the good work, knowing I am not alone in this.

I am indebted to my friends (specifically Steve Wyborney, Kyle Pearce, Robert Kaplinsky, and Kris Childs) who spent valuable time with me before, during, and after the sessions to share ideas, challenge one another’s thinking, and re-commit to the work together.

The sessions I attended each provided a clear, consistent message …

Michelle Rinehart

Math Talks: Adapting the Number Talks Structure for Secondary Mathematics Classrooms

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Christine Newell

Building Mathematical Language and Precision Through Routines

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Grace Kelemanik, Amy Lucenta

Learn How to Develop Teacher Content Knowledge and Practice Through Instructional Routines

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Graham Fletcher

Teaching Beyond the Task: Using Yesterday’s Lesson to Prepare for Today

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Kyle Pearce, Phil Daro

Digging Deep into Ratios and Proportional Relationships in the Middle Grades

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Kristopher Childs

Teaching Mathematics for Social Justice Develops Student Problem Solvers and Not Just Rule Followers

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Shared Albums in Photos with iCloud

The Shared Albums feature within Photos in iCloud provides a simple, systematic way to archive and share images with others.  Any photos or videos on your iPad Camera Roll, or already saved within Photos can be added to a Shared Album.  Though the owner of the Shared Album must have an Apple ID to create the album, there is no Apple ID required to view the album if the owner publishes to a public website (an option within Photos).  Also, sharing is possible with those using a Windows computer, as the images may be housed on a public website that invitees can access via a unique web address.

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Albums can be used to curate photos and videos from invitees, based on certain set up features by the owner.   This minimizes the workflow and necessity to gather images on a single device for a large batch upload of content.  Invitees may also like and comment within the album, should the owner choose to utilize this feature as well.

We photograph things that matter to us.

How might you use Shared Albums to curate and share photos and videos without the barrier of a lengthy workflow process?


Workflow: Archiving and Sharing Paper by FiftyThree Sketchnotes with Shared Albums in Photos

I use the app Paper by FiftyThree to create sketchnotes of professional learning opportunities.   This includes summaries of books, articles, TED Talks, and sessions I attend at conferences.

In order to archive and share these sketchnotes, I use iCloud Photo Sharing.  This allows me to continue to add to the album while sharing a single web address.  I am also able to control the rights of those viewing the images (such as restricting the rights to add images or comments to the folder).

Workflow to Export Sketchnotes from Paper to Shared Albums:

  • While in the Paper by FiftyThree app, view either an entire journal or a single page in the butterfly view.
  • Tap the Export button and select Export Drawings…

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  • Tap on iCloud Photo Sharing.
  • Either select an existing Shared Album or create a New Shared Album.

Workflow to Invite People to Shared Albums on iPad:

  • While in Photos, navigate to the Shared Album.
  • Tap on People and set restrictions (Subscribers Can Post, Public Website, Notifications).
  • I choose to turn off Subscribers can Post and Notifications and turn on Public Website.  This allows me to share the link to the album in iCloud where anyone can view.
  • Share link.

I have utilized other platforms to archive and share my sketchnotes and Shared Albums in Photos is my top choice.

Show Your Work with Pages

The most recent iWork update supports the creation of books and the addition of drawing in Pages with the same ease of workflow that we are accustomed to from Apple.

I put this update to the test by putting it in the hands of first graders and the outcome was fantastic!  The ease of use by 6 year olds was nearly seamless.  The only struggle I noticed was with their desire to change drawing colors.  More on that later.

I gave the first two students (I’ll call them Madison and Eli) math tools of their choosing and asked them to teach me something.

Madison wanted to show what she knew about addition with a rekenrek (the beaded math tool in the picture below).  Eli wanted to prove to me the area of an unfamiliar shape.  Both students followed the same process to create their books.

First, they chose a template.

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Then, they replaced the placeholder on the cover page with their own photo.

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On subsequent pages, they continued to add photos along with annotations using the drawing tool.

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Note for teachers: This is powerful!

The ability for students to annotate their thinking directly on a photo they took provides us with a peek into their mind!  Think of this as the creative equivalent of Show Your Work.  Rather than writing an abstract number sentence separate from the tool (the rekenrek) they use to build it, have the students mark up the actual photo – connecting the number sentence to the tool.

Now, for Eli.  He selected this zig zag shape (shown in the picture below) and said he wanted to tell me how much it was worth.  That’s first grade speak for composite area!

So, just like Madison, he selected a template and began adding photos to replace the placeholders.

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He pointed to the shape and said it was worth 5.  Then, I asked him to prove it.  This is where I am pushing him to justify his thinking – to Show Your Work.

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Without hesitation, Eli grabbed 5 one inch squares, placed them on top of the zig zag shape and took a picture.  Now I know what he is seeing in his mind when he said the shape was worth 5.  He saw 5 orange squares, placed in a non-overlapping way, on the composite figure.  He used the drawing tool to write count the squares and 5.

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The only hiccup in the workflow with these students came when they attempted to change the drawing color.   Each  tapped the color wheel rather than dragging the white teardrop around the circle.  Though this was simple to overcome, it was important because it allowed for the students to choose their own color in order to add their personal touch to their books.

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Another example of creating books in Pages is to capture the process, or tell the story, of a learning experience.  This class had read The Relatives Came by Cynthia Rylant.  The students were tasked with solving the problems the relatives encountered during their visit.

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The image below shows how this student inserted a photo of her cardboard creation (a bed for the relatives) into a book in Pages and began to use the drawing tool to annotate the parts.  She’s writing cup holder on the image.

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This image shows how she added text to the page to justify her thinking – The relatives only had one bed.  There is no need to ask, “What is that? or Why did you make that?” as she has articulated each part of her creation.  As the teacher, I now have a clear view into her mind and don’t need to remind her to Show Your Work.

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I challenge you to use the new books feature in Pages and empower your students to show their work!

The Intersection of UBD & Retrieval Practice

We are making great strides in mathematics teaching and learning.  We are supporting our teachers to understand the content and employ strategies that make the math accessible and rigorous, all while using problems worth solving in class.

Following much conversation, feedback, and contemplation, I am working to design what I am currently referring to as a lesson structures for both my elementary and secondary mathematics teachers.   This comes in response to a request and will be communicated in that way, as opposed to a top down mandate.  Great things are happening in our schools, but our teachers need direction when they are overwhelmed, a nod of support when they are on the right track, and tools for on-boarding when they are new.  That is what this work is about.

Even though we are moving forward in math education, we have an opportunity to collectively do math differently – for the better.  We have the chance to empower our students with skills to retain and retrieve what they have learned so that they may connect, compare, extend, transfer, and create with this knowledge.  The problem is that we do not do this well, if at all.  Why do we not teach our kids how to use metacognitive skills to own their learning?  I cannot grasp why we are not transparent with students, providing them with clarity about what they will learn, what they are learning, and what they have learned so that they can be in control.

So, consider a team of teachers gathered around a planning table, looking at the next unit of study, making decisions about the learning experiences they will provide for their students.  What steps do they take?  Do they use a calendar to plan 8 consecutive lessons followed by a day of review then a unit test?  I so very much hope they do not.  But, since hope is not a strategy, this is where the lesson structures comes in.

Back to the planning scenario.  Based on that given topic, using the lesson structures, the teachers may choose a few inquiry-based tasks, some whole group mini-lessons with small group next steps, and even some days structured solely as small group days.  Each of these experiences should be purposefully chosen, equitably designed, carefully sequenced, all with learning as the goal.  What I strive to see is learning design that pulls from the best-in-the-world content and resources so that all students have the opportunity to learn and be successful.  I also look toward design that is centered around what the students will do, not what the teacher will do.

In my design there are three lesson structures, each including a few portions that are consistent across the various means:

  • Number Sense Routine;
  • Reflection; and
  • Retrieval Practice.

The Number Sense Routine may look like a Number Talk or various other formats such as Which One Doesn’t Belong?, Notice & Wonder, Numberless Word Problems, Would You Rather, Same & Different, Estimation 180, Count Around the Circle, or Clothesline Math.  The point is that there is an opportunity EVERY DAY for students to make sense of mathematics, develop efficient computation strategies, communicate mathematically, and reason and justify solutions.  The emphasis here is on the students doing the math.

Also, EVERY DAY students are provided an opportunity to intentionally reflect on their learning.  This may include posting content to their portfolio, writing in a journal, or responding to a well written prompt with a classmate.  As adults, we reflect on our day in the car during our commute, at the gym while we work out, or at home as we cook dinner.  We think about what worked, what didn’t work, and what we would do differently if given the opportunity.  Let’s move toward providing this reflective time for our students as well.

The final commonality amongst the lesson structures is Retrieval Practice.  My first experience with this tool was when I read Make it Stick.  Since then, I have listened to the 15 podcasts on the subject by the Learning Scientists, explored the content on the Retrieval Practice website and began reading Small Teaching by James M. Lang.  I have only begun in my learning of this under-utilized opportunity to support our students.  It is exciting to think that a few intentional, purposeful moments in our classrooms can empower our students to become owners of their learning – teaching them how to learn, how to study, and how to use this knowledge to be successful.  Why not?

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In addition to the work to incorporate Number Sense Routines, Reflection, and Retrieval Practice into the lesson structures, I am working to communicate the role each of these (along with Inquiry-Based Tasks, Whole Group Mini-Lessons, and Small Group Mathematics) fit into our structure of Understanding by Design.  The clarity on this connection came to me recently and I am comforted in the alignment – when connections are this apparent, I am assured it is the right thing to do for teachers and students.  Within Understanding by Design, we design learning for one of three purposes: acquisition, meaning making, or transfer.  For many years, we have struggled to articulate the distinction among these three outcomes.  Until now.

  • Retrieval Practice is for Acquisition.
  • Number Sense Routines are for Meaning Making, as are Whole Group Mini-Lessons.
  • Inquiry-Based Tasks, Small Group Mathematics, and Reflection are all for Transfer.

The images above depict Elementary Mathematics lesson structures; however, those for Secondary Mathematics appear very, very similar.

I will continue to work to design supports for clarity around each portion of these documents.  In the mean time, I know the intersection of UBD and Retrieval Practice makes sense.

Make It Stick

Recently I read Make it Stick by Peter C. Brown, Henry L. Roediger III, and Mark A. McDaniel.  My sketchnotes from each chapter are below.

Chapter 1: Learning is Misunderstood

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Chapter 2: To Learn, Retrieve

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Chapter 3: Mix Up Your Practice

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Chapter 4: Embrace Difficulties

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Chapter 5: Avoid Illusions of Knowing

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Chapter 6: Get Beyond Learning Styles

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Chapter 7: Increase Your Abilities

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Chapter 8: Make It Stick

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What Does Learning Feel Like?

Recently, I had the opportunity to spend some time in a 5th grade class, promoting thinking, reasoning, and problem solving – as a model for the teacher.  In general, it was evident these kids could calculate solutions to math problems out of context, but once they were faced with a scenario (concrete or written in text), their estimation, problem solving processes, and justification skills were very weak.

I started by posing the question, “What does learning feel like?”  I wondered if they could describe that feeling of meaning making, that very moment that connections are made and clarity is discovered.  What they shared would have better answered the question, “How do you know that you have learned something?”  After acknowledging the difficulty of describing such an abstract concept, we moved on to the next phase, the Stroop Test.

The students accessed timers on their iPads and prepared to measure how long it took to name the color of all of the words projected on the screen.  As a form of the Stroop Test, the challenge is that the words are actually colors themselves, written in an incorrect color font, as shown below.  Here is a link to my slides.  Note: I also created a slide specifically for a student with red-green colorblindness (to the best of my ability).

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After round #1, I asked the students how it felt to complete this task.  Their responses were beautifully said… It was challenging but I knew I could do it, and I had to tell my brain to slow down because it was saying the wrong words.

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After a second round of the test it was time to move to the next phase of the lesson, the numberless word problems.  Here is a link to my slides.

The idea of a numberless word problem is to scaffold the process so that students slow down and seek understanding of the situation before they allow their brain to jump into number crunching mode.  Read more about numberless word problems from Brian Bushart here.

This first problem is scaffolded in this way:

  • Mr. Choate has some trail mix.  He is going to pour some into each bag for his students.
  • Mr. Choate has 12 cups of trail mix.  He is going to pour some into each bag for his students.
  • Mr. Choate has 12 cups of trail mix.  He is going to pour 1/3 cup into each bag for his students.
  • Mr. Choate has 12 cups of trail mix.  He is going to pour 1/3 cup into each bag for his students.  How many bags of trail mix can he make?

Between the slow reveal of the slides, I posed purposeful questions such as:

What are you picturing in your mind when you read this story?

What is in the trail mix are you picturing in your mind?

How small or large is the bag you are picturing for each student?

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I was surprised, though prepared, to notice that nearly all of the students in this class were unable to reason through the problem accurately.  The most common response I saw was 4 bags.

So, I asked the students to show me with their arms what it would look like if they were holding a bowl that contained 12 cups of trail mix.  The response to this varied greatly.  I modeled this visual as well as what it would look like if I were to hold 1/3 cup of trail mix in my hand.  Again, quite a variety of responses on this one.

I took the response of 4 bags and demonstrated (in an imaginary way), scooping out 1, 2, 3, 4 scoops of trail mix and announced that then my bowl should have been empty.  But it wasn’t.  The students went back to their thinking before I moved to another thought process.

If the scoop was 1 cup in size, how many bags could he fill?  They were (somewhat) able to reason that he could fill 12 bags.  So, knowing the scoop is 1/3 cup, should he be able to fill more or less than 12 bags?  This is where the “backward” thinking comes in to play, much like the Stroop Test.  The scoop is getting smaller while the number of bags is getting larger.

The lightbulb moment for most kids came when I sketched 12 boxes on the board, representing 12 cups/scoops.  The students knew each one actually represented 3 bags, because the bags held 1/3 cup each.  So they added, 3 + 3 + 3 , … 12 times and decided he could create 36 bags of trail mix.

All of that was the first numberless word problem.  Slow, intentional, clear problem solving is what is going to make the difference.  I asked the students who else should take the Stroop Test.  Their response – teachers!  Why?  Because they need to know what it feels like to learn.  Wow.

In the debrief of the trail mix problem, I asked now if they could describe what it feels like to learn.  They said it was stressful and challenging and one even mentioned he had to fool his brain and not let his brain fool him.


My next steps include working to incorporate more number sense routines in resources and professional learning.   Some of my educators are using these structures, but all students should have an opportunity to think about number and scale and reasonableness more often.

  • Problem Strings
  • Number Talk
  • Which One Doesn’t Belong?
  • Notice/Wonder
  • Numberless Word Problems
  • Would You Rather?
  • Estimation 180
  • Clothesline Math
  • Count Around the Circle
  • Tools such as number lines, rekenreks, hundreds charts, strip diagrams

The Power of the Ask

In September of 2017, I was asked to return to the 2018 Research-to-Practice Conference at Southern Methodist University.  At the 2017 conference I served on the panel at the opening of the conference.  This time, I was the closing speaker.  The conference is exceptional – a best kept secret that sold out this year in 48 hours!  Always working to improve upon my practices and knowing the structure of a full 60-minute time slot, a given topic problem solving, and the final event on a Friday afternoon, the pressure was on.

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As with all of my planning, I employed backward design, beginning by making notes about the biggest takeaways I wanted to provide for the audience: actionable content grounded in equitable practices.   I built upon what I know about breaking barriers to accessing mathematics and quickly recognized that my network of math education leaders have made significant contributions to this work.  I wanted the audience to be afforded the opportunity to make these connections as well.

Enter: The Power of the Ask.

I drafted personal emails to nine of my friends, requesting their insight on this topic.  This is what I sent:

I have the opportunity to present the closing session at the 2018 Southern Methodist University, Research-to-Practice Conference in February (link).  My topic is: Breaking the Problem Solving Barrier (description below).  Info from last year’s conference: here.
Too often the problem solving tasks we design include unintentional barriers that prevent our learners from accessing the mathematics. Language, prerequisite knowledge, disengagement, limits, and time are five barriers impeding learning in our classrooms. Let’s break the problem solving barrier in order to cultivate a problem solving community in which all are welcome.
I am working to design my content around the expected audience of approximately 300 mathematics educators and teacher leaders.  I would like to send them off with both practical content and the reasoning behind the benefits of removing barriers in mathematics in terms of problem solving.  Even better, I would love the opportunity for these teachers to hear the voices of many leaders in math education who are collectively working to make the content accessible for all students, highlighting their favorite structures or tasks.
Here’s where you come in… I would be so grateful if you would contribute to this FlipGrid Topic, sharing your favorite problem [structure, task, lesson, system] that allows the mathematics to be accessible to learners.  Don’t be too humble to toot your own horn in sharing something you designed, or give a shoutout to a colleague for the work he/she has done!  I greatly appreciate your contribution and know the mathematics educators and teacher leaders will benefit from your words of wisdom!
Wondering what to share?  [This is where I added specific work each of my friends has created that would be valuable for the audience to see.]
What happened next honestly surprised me.  Every one of them responded YES!  Every one of them contributed to the FlipGrid topic I created.  Not one of them was too busy, not interested, or didn’t see the value.  Every one wanted to (voluntarily) support teachers and instructional leaders for the greater good.  All I had to do was ask.

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I was able to share the voices of my friends Brian Bushart, Graham Fletcher, Cathy Yenca, Steve Wyborney, Jon Orr, Kyle Pearce, Lucy Grimmett, Michelle Rinehart, and Robert Kaplinsky with those teachers and instructional leaders at the conference.  I was able to send them off with actionable content grounded in equitable practices.

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My biggest takeaway from this experience is twofold:

  1. Continue to ask.  We are better together.   Also, being humble enough to ask others to fulfill what you need will allow you to grow in ways you cannot anticipate.
  2. Continue to say yes.  When others approach you with a need, say yes.  You will grow by giving to others.

Have you seen Jia Jiang’s TED Talk What I Learned from 100 Days of Rejection?  Worth the watch if you wonder about the power of the ask.

Why Does Math Look Different?

The most common question I am asked related to my profession (by adults mostly outside of education) is, ” Why does math look different than when I was in school?”  If I was speaking to you in person, I would share my thinking with lots of gestures and excitement because I love math that much.  In lieu of a personal conversation, I will do my best to paint the picture with words.

The short answer: It’s you, it’s not me.  The long answer: It’s all about perspective, and here’s why… Constructivism is rooted in the work of Jean Piaget from nearly 90 years ago.  This theory of learning best describes the phenomena (to many) that is modern mathematics teaching and learning.

At any given point in a student’s learning journey, he/she has some accumulation of mathematical understanding.  Then, for every learning experience, opportunity to do math and reflect on the thinking behind the calculations and representations, the student’s networks of understanding are added to, changed, or somehow extended.  This is how you know learning has occurred – it has impacted the student’s thinking either through verifying beliefs or challenging understandings.

Now, when the students move along this continuum of mathematical understandings, they strategically modify or verify their existing knowledge or they abandon it for more efficient, accurate methodologies or connections.  This is powerful – strategic abandonment of ideas for new, more relevant ones.

Back to the perspective claim.  As adults, we see the methodologies of teaching and learning mathematics from our current state.  We are efficient, accurate mathematicians.  We know what tools to select to solve problems, appropriate processes to use to solve problems, how to verify accuracy of solutions, and means to explain our thinking.  We stand very far down the mathematical journey, relative to the position of children in elementary or middle school, for example.  If we were to walk ourselves back down this path, past when we first used a calculator to balance our bank accounts, past when we first learned Algebra, beyond our days of fractions and long division, we would find ourselves in a world of inefficient mathematics.  But this is all relative.  This is about perspective.  Using repeated addition, such as 4 + 4 + 4 + 4 + 4 + 4 + 4 to arrive at 28 is less efficient than using multiplication, such as 4 x 7 instead.  And, counting 1, 2, 3, 4 and 1, 2, 3, 4 and 1, 2, 3, 4 (seven times) is even less efficient.  Once we truly understand that multiplication is a more efficient method to find the total number of objects in a given number of equal groups, we strategically abandon the method of repeated addition.

So, math does look different.  This is true.  However, it is all about perspective.  You should change your perspective and put yourself in the shoes of the children.  You were once there, but you have strategically abandoned those memories for ones that are more efficient!

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Change your perspective and you may understand why math looks different.


Want to learn more?  I recommend Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, and Bay-Williams.

The Power of Constraints: Binary Questions

One strategy I often employ in professional learning with my educators is to rate themselves on a scale of X to XX.  Similar to estimation on a number line, they are to reflect and consider where they stand with relation to a given prompt.  Though this line of thinking supports “correct” responses anywhere along the continuum, it is missing one key ingredient: a constraint.  There is no true decision making required, no absolute committment.  There is no one or the other.  No A or B.  No 1 or 0.

Enter: Binary Questions.

Recently, I introduced the concept of Binary Questions to my educators in a professional learning experience.  This small act prompted many, many conversations centered around possible prompts for their own classroom.  (Success!).

The constraint of decision making adds a twist to this reflection in a new way.  This is not a popularity contest or true/false question.  Rather, it is a visible sign of individual voice.

In the future I will not abandon the number line, approximation form of questioning by any means, but I will certainly consider this structure as well.

I am building a list of Binary Questions and archiving them here: https://www.icloud.com/numbers/0jZb9r3CqPuI1B6Es1OuPmIZA#Binary_QuestionsScreen Shot 2018-01-25 at 7.35.50 PM