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 , , , | 2 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.

FullSizeRender

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.

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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…

FullSizeRender-3

NOT A STARTING POINT!

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.

FullSizeRender-1

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…

FullSizeRender-6

Invert and multiply.

 

 

 

Posted in 3-5, 6-8, Against the Norm, Fractions, Making Sense Series, Math Progressions, Number Sense, Planning, Strategy Development | Tagged , , , , , , | 12 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 | 7 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

Becoming a Better Storyteller

I recently had the amazing opportunity to be 1 of 6 speakers at ShadowCon16.  The whole premise of ShadowCon is awesome and full credit goes to Zak Champagne, Mike Flynn, and Dan Meyer for having the vision and bringing this all together.

Conferences quickly become distant memories, however our three friends have found a way to extend the conference experience. Each of the six ShadowCon speakers shared a provocative 10-minute talk and then challenged the audience (YOU) with a Call to Action.

You can find my ShadowCon Talk here, along with my call to action and some extra tidbits. After watching my talk, please take the opportunity to respond to my call to action by posting your thoughts on the message board.

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If my talk didn’t resonate with you and you don’t feel compelled to act…no worries because there 5 more chances to get connected.

Robert Kaplinsky – Empower

Gail Burrill – Math is Awesome: Let’s Teach so Our Students Get It

Kaneka Turner – Extending the Invitation to Be Good at Math

Brian Bushart – Make Your Own Kind of Music

Rochelle Gutierrrez – Stand Up for Students

I have purposefully closed comments for this post in hope that you will continue the conversation on the webpages that NCTM has graciously shared.

I look forward to learning and growing together as all of us are smarter than one of us.

 

 

Posted in Against the Norm

Be the Teacher: Moving from Counting to Cardinality

Cardinality…what is it and what does it look like?

If you’re not a kindergarten teacher you might be left shrugging your shoulders if someone asked you to define cardinality. Before students can own the idea of cardinality, they need to have an understanding of one-to-one counting.

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So now you know what it is, here’s an example of a 4-year-old working to gain an understanding of cardinality…

 

If you’re not familiar with cardinality you expect this little guy to say “seven”. Especially after he’s counted and I’ve repeatedly ask him “how many are there?”

But nope, he’s not quite ready.

Be the teacher… 

What activities and/or suggestions can you offer to help me move this little guy from 1-to-1 counting to cardinality?

Please share you thoughts in the comments.  My hope is that by sharing this video parents can begin to look for and promote cardinality at home before students get to kindergarten.

 

Posted in Against the Norm | 41 Comments

The Progression of Addition and Subtraction

The more I create… the more I learn.  Here is the 3rd installment of this whole Making Sense Series which has truly forced me to be a better teacher.  A more educated teacher.

I can’t stress enough how much I’ve learned from diving into the progressions found here. There are many intricate pieces of learning which are mentioned throughout the progression documents which can’t be included within a 5 minute video.  I strongly recommend finding the little pieces with your team or grade level.

Unlike the multiplication and division videos which were introduced separately, this video is an addition/subtraction mash up…they way they should be explored by students.

Posted in Against the Norm, counting, K-2, Making Math Accessible, Making Sense Series, Math Progressions, Number Sense | 25 Comments

I’m Placing a Hit on the Pseudo-Context

This robs students of everything mathematics should be…

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  • The “real world” connection
  • The step-by-step procedure
  • Circle the numbers you need to solve the problem
  • The pseudo-context word problem.
  • Lesson 19.2 infers that this unit is front-end load with procedures and formulas

What a sham!

I can’t help but ask the question FOR students, “When will they ever use this?

As if that wasn’t enough, the “real world” problem is followed up with mind-numbing practice…

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Notice that question #7 has been labelled as “H.O.T.”

Instead of going on a rant, I’ll keep my head down and continue to do my best providing alternative opportunities for students and teachers to engage in math.  The Fish Tank and Got Cubes are two examples that push back on pseudo-context questions and volume.

As for Mike’s DVD box, H.O.T question, and curriculum developers, I’ll offer this new task as a means to undo the limited understanding you promote with every edition you pump out.

Act-1

Question: How are the sugar cubes packed in the box?

There’s no empty space.


Act-2

Number of Cubes in the Box

I’ll let students play around with the dimensions for a while. Then I’ll share this video of me putting in the last cube.

 

Now students have two dimensions and know the total number of cubes.


Act-3

How the sugar is packed


As a potential extension I might ask if this the most cost efficient way to box the 198 sugar cubes. Sure it dives into middle grade standards but it seems like a natural progression.

I doesn’t seem forced and that’s what I’m after.

Posted in 3-5, 3-Act Tasks, 6-8, Against the Norm, Cheese Mover, Intellectual Need, Measurement and Data, Modeling | 20 Comments