Mathematical Goals and Geometry and WODB

(Full disclosure: I was totally inspired for this lesson by Tracy Zager [as I often am] and her recent blog post Straight but Wiggled in which she uses this same image.)



Today was one of those math conversations that just got us all pumped up!

To talk about today's lesson, I have to start at yesterday.  Or even several weeks ago, with a conversation with Jamie Duncan about planning with mathematical goals in mind.  She tweeted to me this image yesterday.  

I had this immediate feeling of frustration and disappointment in that day's geometry lesson.  Although students really enjoyed using the different tools available, I felt like we were missing depth to our work.  They made three sided shapes.  Called them a triangle.  Missing depth for sure.  We were missing the mathematical goal.  I knew I wanted and needed to do something different for the next day.  

Missing depth!

Today we started with this Which One Doesn't Belong image:



Usually when I do WODB I ask students to pick an image, go to the matching corner of the room and talk with the other classmates who are there about why it doesn't belong.  Kids liked that, but it didn't allow for any discussion: they are in the same corner for the same reasons, mostly!  Today I decided to keep everyone on the carpet together, and boy am I glad I did!

After displaying this on the SmartBoard, out of the corner of my eye, I saw my most nervous mathematician raise his hand quickly and then put it right down again.  I can imagine his brain thinking "I have some ideas but hellllllll no am I going to share them!".  I've been trying all year to boost his confidence in math in a really authentic way.  He has some big holes in his math understandings, but I want him to feel confident in the things he does know.  Because he knows a lot.  So I went over to whisper and ask his ideas.  I told him I was going to call on him and I did it immediately.  

He shared this idea: "[the top right] doesn't belong because it's black".  But he wanted to go on! "[The bottom right] doesn't belong because it has 5".  Another student chimes in "five what?" and he confidently answers "5 sides".  This is a low floor task and he could enter and stay inside.  He was hooked!

The conversation quickly went to looking at the top left corner.  One student said "[the top left] doesn't belong because the others are fat and this one is thin".  Immediately another student said "I want to add to his ideas".  He added "because of the short lines".  I pushed him to explain how his idea actually relates to his classmates' idea.  This led to another student saying that the lines are different sizes on that top left shape.  

This was the first time anyone had mentioned anything about the size of the sides rather than the amount of them.  This led to some classmates disagreeing and saying that the lines on the pentagon were also different sizes.  This led to more disagreement and we figured out how to find out if they were the same size or not.  We took out the ruler on the SmartBoard, but couldn't manipulate it to measure those top diagonal lines.  So another student mentioned that we could trace each line with the SmartBoard marker and then move it on top of the other lines.  Wow.  Minds were blown.  The sides of the pentagon are in fact the same size!

Energy level was high! Kids are pumped!

But the conversation goes on!

We end up talking about the bottom left shape.  By this time we have lost all form of organization and control.  Some students are sharing out their thinking while others are desperately waving their hands in front of me to be called on!! {Side note: this is a challenging moment for me.  I want students to be able to just free flow with their ideas, but some students are just not into that.  And if we're all free flowing, we can't listen to each other's thinking! And we typically agree to raise our hands to talk.  I just love when a conversation is flowing! I don't want to stop it to call on kids-but I know it's not fair for those kids who are following our agreements!}

We talk next about how the bottom left shape is a diamond.  Rhombus?  Is it a square? What if I turn it a bit? Is it a square now?  What does that do to the other shapes? Does this change their names?

I finally had to end the conversation.  I heard "It's not fair!" and "Awww!".  I love that.  I want to end the conversation right at that sweet spot.  Not once they are bored and disengaged anyway!

This was our final image from our conversation:

Doesn't quite capture the depth nor emotion of the conversation!

Their task for today was to create shapes (triangle, quadrilateral, pentagon and hexagon) with pattern blocks and tangrams.  They had to use at least two shapes to create another.  I was hoping for some creativity!

I discovered that the grand majority of my students (all of them I might venture to say) think that it's only a shape if the angles are convex [my word, not theirs].  I got a lot of shapes like this:

I talked to one student who was just pulling her hair out trying to make a hexagon.  I caught her with the hexagon she currently has here in this photo:

I asked why she couldn't use that one.  She didn't think it was a hexagon!

I talked with another student who had created a shape similar to this:

She called it a hexagon.  See why?  She only counted the four "corners" (as they were taught in first grade) and the two tips of the triangles.  When I asked how many sides it has, she said 10.  

Great conversations.  Great understandings and great misconceptions.

Takeaways: 

-Plan my math lessons with the mathematical goal in mind.  We won't have as many shallow lessons. 

-Use activities like this as pre-assessments and information gathering.  I glean so much more than when using a paper and pencil assessment.  (Need a better way to keep track of what I learn about individual students, for now I just have information about my class in general.) 

-Keep reading other people's blogs.  I always get amazing ideas from them. 

-Keep using and promoting low floor, high ceiling tasks.  They let everyone enjoy the math!

-I need to show the same enthusiasm for read aloud as I do for our math talks.  Those are really falling flat.  Any tips?!

Boy does it feel good to get these ideas out of my very busy brain. 

3 Act Reflections

I just watched Kristin Gray's Ignite about Creating a Culture of Learning and it inspired me to do a little of my own reflecting.  Don't get me wrong-I'm always reflecting.  Stewing, perhaps, would be a better word.  But yeah, let's call if reflecting.  Sometimes I feel the need to do it more productively than just going over and over it in my mind.   So here goes.

Last week we did one of Graham Fletcher's 3 Act Tasks called Downsizing Tomatoes (a task to which I have a special bond via a ketchup bottle).  Basically Graham pours "ketchup" out of a larger bottle into smaller bottles.  The question is, how many bottles can he fill?  We are knee deep in subtraction and we needed to shake something up a little.  Somehow we had found ourselves in a rut and I needed to dig us out! 

We hadn't done a 3 act in a while and I knew the problem was going to be hard.  I planned to give us a couple of days to work on it, but that was probably the extent of the planning that I did.  Routines are pretty solid in my class: we know how to notice and wonder, we know how to estimate, and we know where to find tools.  I knew it'd be challenging, but I was totally confident that we could do it.

Well, results were mixed, as expected.

Firstly, students were really bent out of shape that the liquid inside the bottle was not actually ketchup.  They noticed it was honey, maple syrup or apple juice.  They just couldn't believe that it was water and food coloring.  We decided to call it fake ketchup. They could deal with that. 

I also had a lot of students who noticed that there were 10 small bottles and that really got in the way when they were trying to figure out how many little bottles were filled.  It comes down to many students not knowing what question they were solving.  

This student, for example, really persevered and found out how much ketchup Mr. Fletcher would need to fill up all of the 10 bottles.  Great strategy, nice use of base 10 blocks...not the question we were answering! (We have the information to solve this problem, but considering I wanted us to have the potential for subtraction, this was not the question we were going to use.)

Then we have this students' work.  She was making groups of 64 on the 100s chart.  She seemed to have made sense of the problem and certainly persevered in solving it.  But she has some scribbled out numbers that apparently don't count, and only made it to 300, instead of 397. (Upon revision of her work, she realized her mistake.)



Then we have this friend, who again was solving the wrong problem by focusing on the 10 bottles.  But when he got his answer, he did not put it back into the context of the problem to see if it made sense!  164 bottles?! (And yes, we need to clarify a bit on this repeated addition and multiplication stuff!)



I also had some students who clearly made sense of the problem, but didn't quite know how to use the math to get to the right answer.  



Some students were being very very creative with their tools-good thinking, but I'm not sure our balance scales are accurate enough to solve the problem like this!

Ok, it's not all bad.  I also have this student who revised her work THREE TIMES before she got the right answer.  She was so thrilled to do it! Talk about perseverance!  (Although I'm still not sure what strategy she used for all that addition!)



So after the second day, I felt super frustrated.  I still had a significant group of students who were just not making sense of the problem and determining the right tool to use.  After some conferring with Jamie Duncan I decided that we needed to go back to the basics.  I had a group of students to whom I gave some specific feedback using seesaw (my miracle tool!) who I thought could realize and revise their mistakes on their own.  And another group that needed to see 397 ones blocks in a jar get spilled out, 64 at a time, into smaller jars, replicating exactly what Mr. Fletcher does in the video.  Students put the 64 ones blocks into the cups.  Some students started working with money and realized they needed to do some decomposing! Students counted and recounted.  Students engaged in mathematical discourse.  We even talked about precision when students said a half of a jar was filled, instead of just a little more.  All good stuff!


We finally were able to share Act 3, students felt good about their work (regardless of whether they had the right answer or not), and all of our brains hurt! 


I have some takeaways:
-this group of students needs to make sure that they think about what tools they want to use and how they are going to use them before they even start.  
-students should write down in their notebook what problem they are solving to avoid frustration later!
-some of my students are still very concrete learners.  I need to plan better for this.  
-I need to solve the problem myself.  I know this-unfortunately the time crunch gets in the way (excuses, excuses).

We are going to do act 4 tomorrow.  I found a jar of Chili Sauce in my fridge that I emptied out (I think it was getting moldy anyway!) and will fill with water and food coloring.  I will pour it into a smaller jar, just like Graham does.  I want to see what students will do this time.  Will they go for the same strategy? Will they use a classmates strategy? Will they request the bottle to do the pouring themselves?  Will they learn from their past mistakes?

I'm also debating with bringing a Franks Red Hot bottle also. It's much bigger.  A couple of my students who are more experienced with addition and subtraction might enjoy a tougher challenge.  Or, maybe I just change up the question to have an open middle: {still musing on this question}.