Colorblind Teachers, Invisible Students (Ignite – GCTM 2016)

We recently finished up our state math conference here in Georgia. Last year I shared our Ignite Talks in this space and plan to do the same with our 2016 session.  As I edit the videos and prepare to release them, there’s one talk I’ve watched multiple times.

Carla Bidwell’s talk was a gut check and really spoke to me as a white educator. I hope it does the same for you.

All of us are smarter than one of us.

Posted in Against the Norm | 1 Comment

The One-Handed Clock in a Digital Era

It’s tough to make a case for the analog clock in our digital world. I’ll leave the debate of its relevance up to the professionals. Nonetheless, the analog clock remains a staple in the majority of math curriculums.

Let’s take the following standard for example:

MCC2.MD.7 – Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

Screen Shot 2016-10-31 at 8.04.13 AM.png

One could say that this standard applied to a digital clock inadvertently supports the work of rounding in 3rd grade.  8:04 is nearest to 8:05.

But in the same breath, I can hear the conversations taking place when students try to apply the same understanding to the analogue clock.

Screen Shot 2016-10-31 at 8.08.04 AM.png

“Which one is the minute hand? The long one is the hour right? No wait. Which one is the hour hand?”

What if we asked students to estimate the time with the hour hand and completely ignore the minute hand? Then, once they own the estimation of time using the hour hand we introduce the precision of time using the minute hand?

This is not a new idea and I’m definitely not the genius that thought of it. Patricia Smith and John Van de Walle have both tackled the issue of time well before this post.  I just figure the more that know the better.

So let’s give it a try…About what time is it?

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Time we go to specials.

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Recess time.

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Time to go home.

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Bed time.

A friend in my district reminded me of a conversation we had last year about using a one handed clock and it’s almost that time of year for 2nd and 3rd grade.

Share these pictures with your class and ask “What do you notice?”  Turn them into some kind of Time Talk and report back. I’d love to hear how it goes.

Does EVERY student need this clock? Not at all. It’s just another way to make learning accessible.  Remember what Dan said “You can always add to a lesson, you can’t subtract.

The same applies for clocks as well.

Posted in Against the Norm | 5 Comments

3-Act Task: A kindergarten lesson captured

Over the past few months I’ve been asked for videos that capture a 3-Act Task being taught in the elementary grades. I didn’t have any, or know of anyone that has captured an elementary 3-Act except for this Teaching Channel piece.

Before moving forward, this post wouldn’t be possible without Dan’s trailblazing skills and introducing us all to 3-Act Tasks.

Last week while visiting a kindergarten class we tackled the Candyman. So in the spirit of vulnerability, here it is and I’ll take whatever feedback you can offer.

Act 1: Notice and Wonder

***The blur will disappear after 30 seconds. Little man had to blow his nose***

Act 2: Identify Variables and Solve Conflict

Here’s the kindergarten recording sheet our friends used to make estimates and show their thinking.

After students estimated, we shared and identified the variables needed to answer the question, “How many candies were in Mr. Fletcher’s hand?”

At this point students were good to go and got their model on.  We didn’t scaffold learning in any way because we were using this as a formative assessment.

Here’s what we got…

The context and colors of the candies really helped students explain and model their thinking.

Not all was gravy. There’s still lots of work to go but that’s to be expected. It’s kindergarten and we’re in October.


This student has mastered the art of drawing zeros. Lots of them.


Others looked rough too…


But when we looked closer and talked with the student, they counted the square pencil boxes for us. Awesome to uncover this hidden gem.

Really surprised to see this…


But my mind was blown with this little guy…


10 was his answer but 8 is what he modeled.

So we wanted to know more…

Mic drop.

Act 3: The Reveal

It’s always great to engage the youngins’ in 3-Act Tasks. I’ve heard colleagues say, “I don’t have time to do these types of lessons.”

I hope this helps in showing that we don’t have time, to not have the time.

Thoughts and feedback welcome.

Posted in Against the Norm | 25 Comments

Geometric Subitizing: A Different Kind of Number Talk

These Geometric Subitizing Cards have been in beta the past two years.


Geometric Subitizing Cards

I asked our buddy Joe Schwartz to take them for a test drive last spring and he reported back.  Joe’s all in.

How do they work?  Let’s play…

Last week I took them for another spin in some kindergarten classes. Here’s two examples of how it went and we’re only 2 months into the school year.

Last year I captured me playing cards with my girls.  Here’s me playing with my kindergartner. 

Here’s me playing with my 3rd grader.

Your turn.

Take these cards and give them a try. If you need some ideas how to use them you might want to check out this task card.  I plan to share my thoughts about Geometric Subitizing in a future post but for now I just want you to go and play. The only thing I ask is that you report back and offer feedback.

  • How did it go?
  • How can we make them better?
  • Any other ways we could use them?
  • What am I missing?


Posted in 3-5, Against the Norm, Algebra, counting, Fact Fluency, Making Math Accessible, Math Tools, Number Sense, Strategy Development | 26 Comments

The Task Graveyard

It’s where my ideas go to die.  But like the mighty phoenix, I have one idea for a task that keeps rising from the ashes. It just won’t die.

If Bruce Willis was a math task, I’ve seen his face.

I can’t help it. I’m fascinated by them. The containers below have sat on my desk for almost a year and everyday they taunt me.  Like a banshee in the night, they scream to be played with and I couldn’t take it anymore.  So yesterday… WE DANCED! 

Now I’m clueless. Where do I go from here? 

Before I rebury this guy in the file “Never to be Found Again,” I would love for you to play #WCYDWT with me.

I’d love to hear your thoughts, ideas, where you think it fits (grade level), or whatever else you could do with it.

For what it’s worth…here’s what I know and where I’m at so far.

Posted in Against the Norm, Intellectual Need, Making Math Accessible, Modeling, Planning, SMPs, Teacher Content, Teaching in a Context | 7 Comments

Conservation: The more things change, the more they stay the same

Every once in a while I stumble across Piaget’s 7 Conservation Tasks and then I move on. This week was different because my thoughts around conservation won’t go away.

In a recent meeting with kindergarten and 1st grade teachers we discussed the importance of students being able to decompose and unitize number. This meeting has left me thinking that conservation is an conceptual underpinning of both skills. But I could be wrong.

In an effort to explore conservation we shared some videos in K-1 classrooms around our district.

We started here…



We paused the video at what do you notice and we shared our thoughts as a class.

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Gotta love what Kindergarten students notice.


Right before the flattened clay was placed back on the scale we hit pause again…

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Students predicted if the clay would weigh more, less, or the same using a slip of paper and unifix cubes (below).  With this being a kindergarten class we spent time talking about letter sounds at the beginning of each word and what each word meant.

Some students struggled with the letters/sounds but the majority of students were able to identify where more, less, and the same were located.  This turned out to be a quick and efficient way to formatively assess in multiple areas.

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We gave students an opportunity to share and explain their predictions.  Sharing predictions was the easy part. Explaining and justifying our predictions, not so much.

So we played some more and followed up with this…


And this…


And finally this one…


The whole idea of unitizing is a foundational understanding that students should own by the time they leave the primary grades. It’s our hope that by introducing the concept of conservation as it relates to length, weight, and liquid, that students are able to make connections across different contexts.

We hope that these contextual connections begin to help students explore how 1 ten is the same as 10 ones.


But it’s the fourth week of school and we know we have our work cut out for us.



Posted in counting, Estimation, K-2, Making Math Accessible, Strategy Development, Teacher Content, Teaching in a Context | Tagged , , , | 5 Comments

Building a Conference

I’m currently serving on the 2017 Orlando Regional Program Committee and honored to work with some great folks. When we met over the summer there was one thing we knew right from the start, we want this conference to be different.

In our attempt to be different we asked for, and were granted a #NCTMregional blog. We hope this helps the communication flow both ways.

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Over the next year we will post upcoming release dates and solicit your feedback as we look to build a conference that  “will reflect and represent the diversity of our profession. As we look to build a conference that is inclusive, we welcome, value, and need your input.”  (NCTM Regional Blog)

As we begin this journey together, please take a moment to complete the 3 question survey found on your Orlando Regional Blog. What do you want or expect from an opening session at a NCTM Regional Conference?


Comments are closed for this post. Please share thoughts and feedback in the comments at the blog.

Posted in Against the Norm

Making Sense of Invert and Multiply

As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction.  When we do, it’s normally accompanied with Keep-Change-Flip or the saying “Yours is not the reason why, just invert and multiply.”

Both are conceptual cripplers.

I’ve been drafting the 4th installment of the Making Sense Series involving fractions and I’m sharing this post as more of a personal reference should K-C-F make its way round these parts again…and I’m sure it will.

Side note: A while back Fawn and Christopher each shared a post about division of fractions using common denominators. Both posts left lots of math residue and are well worth your time.

Let’s start with a model for 3/5 ÷ 1/4.


Modeling measurement division of fraction by a fraction.


At some point along the way it becomes inefficient for students to draw models once the conceptual understanding is established. As students represent measurement division of fractions they should be formally recording their thinking.



From here students generate their own algorithm (shortcut). They begin to recognize that they will always get a denominator of “1 whole” so they begin to purposefully leave it out.  In doing so, they become more efficient in the procedure of dividing fractions.

Some students begin to eliminate the green and red steps from the above equation because they’re seen as repetitive.  We’ve even had one student that “invented” and generalized cross multiplication for division of fractions as they searched for ways to record fewer numbers and symbols.

It looked something like this…



I keep reminding myself that if fractions are the gatekeeper to algebraic reasoning then I need to slow the process down and conceptually understand what’s happening. This includes K-C-F.

As students understand the power of creating a whole number denominator they begin to search for more efficient ways to get 1 whole.  It’s here that they being to explore with equivalent fraction and the idea of using the reciprocal to create a whole.


What I’ve realized is that aside from complex fractions, the underpinnings of this equation are developed in the elementary grades.

  • 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a÷b = a/b).
  • 4.NF.1-Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b).
  • 3.NF.3.c- Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Just like before, students will look to generalize the equation above and find shortcuts. They’ll do so by eliminating repetitive steps which would leave them with…


Invert and multiply.




Posted in 3-5, 6-8, Against the Norm, Fractions, Making Sense Series, Math Progressions, Number Sense, Planning, Strategy Development | Tagged , , , , , , | 15 Comments

The Multiplication Sundae and the Bad Taste of Incentives

Earlier this week I received an email asking for incentive ideas for a school wide fact fluency focus:

Hi Graham,
I need some insight about math facts and incentive programs. I follow your site and have read Not Your Mom’s Flashcards:Conceptual Understanding of Multiplication and watched From Memory to Memorization: There is a Difference. I know facts are about being efficient, accurate and flexible. With that said our school would like to start an incentive program to encourage students to learn their facts. Ex. In the hallway, picture of student and medals earned and a quarterly bingo math party with snacks and prizes. I am all about balance but… I want to make sure we do what is right for all students. My principal would like an incentive program for facts and I want to lead us in the right direction.

If you search the Internet, there are tons of incentives for fact fluency and the Multiplication Sundae is a big seller.  But the problem I have with the sundae is that some kids never even earn the bowl, let alone the ice cream. And the cherry? It doesn’t stand a chance!

In the same week, I was reminded why incentives for fact fluency crush my soul.  I was at my daughter’s award ceremony, she’s a 3rd grade student. During the presentation they awarded all students that had mastered their multiplication facts with an award. There were a handful of students from her class that earned this award. As a dad, I was proud because my daughter received the award but I know she learned her facts the right way.  But what absolutely crushed me is the other 17 students in her class that didn’t receive the award and how they now believe they’re not good at math.  So I’ll ask the question…Is the award worth it?

I really appreciate the email and all the work we do as teachers to motivate our students but now I can’t escape 2 questions:

  1. Is there an incentive idea/program that addresses equity? An idea where EVERY student can be successful?
  2. What role (if any) should incentives play in our schools?

I like to think if we can’t address question #1 with “yes” then question #2 is answered for us…incentives don’t belong.


Posted in Against the Norm | 10 Comments

3-Act Tasks (#49, #50, #51)

They come in waves. It’s both a blessing and a curse.

Seems like everywhere I walk there’s a perplexing question I can’t just let go of. The bug of mathematization has hit my entire family.

My wife is a teacher and for appreciation week she was given a light bulb filled with Skittles. She comes home and says “Hey Gray, I’ve got a 3-Act Task for you!”

And that’s how task #49-Bright Idea came to be.

We love pickles and jalapeños in our house. We have back-ups for the back-ups. My daughter in 3rd grade was analyzing the jars while unpacking the groceries. Then she says, “I wonder how much longer it takes to fill up the little jar compared to the big jar?”

Enter task #50-Dill ‘er Up

I was at a friend’s house and he was slicing some apples for our girls to have as a snack. My youngest daughter (kindergarten) watched intently and ask “Hey daddy, how long do the think the skin will be? Will it be taller than you?”

Voila…task #51-Granny Smith’s Skins


In September 2013 Dan visited my district and said I needed to start sharing and I haven’t stopped since.

51 tasks later and I just want to say thanks.

  • Thanks to my family for putting up with me. Sure it’s fun but I know it’s annoying too.
  • Thanks to Dan for the push to share.  I hope my work has inspired others to share openly and freely as well.
  • Thanks to my friends in the #MTBoS who help push and question me. With late nights and behind the scenes magic… you make me feel normal.
  • Thank YOU for your continued support and encouragement. If you’ve used or shared one task, I appreciate you drinking the Kool-Aid. Thanks for the feedback and for sharing pictures of student work. You inspire me to create more.

“All of us are smarter than one of us”


Posted in Against the Norm | 12 Comments