Practice Puzzles: Math Practice Three

Math Practice One and an explanation of the puzzles
Math Practice Two

There is nothing more charming (and alarming) to me than to listen to a child try to explain themselves mathematically. I truly do love every second of it...I never tire of it, and it never gets old. Having said that, I'm not the only teacher who has to reconcile the balance between pushing past, and lingering in, the nonsense. In all of these Math Practice Puzzles that I am sharing, there is way more to the practice than I have given rise to here in these captured moments. The point was to find ONE thing we could use as an entrance to the practice...I wasn't looking to frontload it in its entirety, I just wanted one shared, contextualized experience to draw out the practice. The rest has unfolded as it has, with some practices more easily expanded on than others. Which feels exactly as it should for working with the Littles. They have Kindergarten until Senior Year to sort it all out. I'm just doing my small part.

So, Practice Three....there are so many deep and important truths in this practice, but the one that we have cleaved to early on is the idea of revision. This practice is tightly wound with other practices, and as we see more patterns, learn more structures, take command of more precise language, and test more solutions and strategies, our communication and explanations will continue to evolve. But these are lofty aspirations indeed, if you consider how we start on this path. There is no primary teacher on the planet who hasn't heard such carefully crafted nuggets as: "I know it's five because that is my favorite number." Or, "I know because I knew it in my head" (which is MOST exciting when it comes with a wrong answer...."8 + 3 =12, because I knew it in my head"). Other oldies but goodies include "I counted on my fingers" and "I guessed".

Early on, I introduced the notion that the onus is on the explainer....if somebody doesn't understand you, it's your job to keep revising your explanation until they do. It's not because they aren't listening to you (they aren't though) and it's not because they aren't smart (they really are) it's because the explanation wasn't sufficient. The correlating piece to this is it doesn't have to be perfect to get started. Focusing on revision means that all you have to do is start. Just say SOMETHING. Then you'll get some feedback, and you can add on/change/delete to make it more clear. Get more feedback. Make more changes. And so on. You don't have to wait until you know exactly what to say or how to perfectly explain something. Just say anything.

From the classroom: Afoa had JUST told us that the two numbers we were going to use to solve our problem were written on the poster, 37 and 43. I turned to Natalia and asked, "So...which numbers do we use?" And she says, I swear, "I don't know." I want to cry haha but instead I tell Afoa, "Well, I'm afraid that explanation wasn't enough for her to understand" and bless his pea-picking-heart he immediately updates it to, "It's the purple numbers on the bottom, 37 and 43." I made SUCH a big deal about this to the whole class! "HOLD UP!! Did everyone SEE what Afoa JUST did???" and then I explained that what he had done was soooooo mathematical because he didn't tell Natalia that she wasn't listening, he didn't tell her oh well you just don't get it, he CHANGED what he said to make it a BETTER explanation." Then Afoa talked about what he had said before and how he changed it. Natalia confirmed that NOW she understands what to do.

Look. It was only marginally mathematical and even calling it an "explanation" is a stretch (it was more like a clarification) but it was all we needed. We have talked about this idea over and over and, in conjunction with a few other key moves, my second graders have made enormous leaps in their ability to explain and question each other. I'm taking it!

Practice Puzzles: Math Practice Two

Math Practice One and explanation of the puzzles can be found here.
 

Math Practice Two: Reason Abstractly and Quantitatively
This practice is when a mathematician makes sense of the problem in context, then ignores the context to do the calculation, then steps back into the context to make sure the answer makes sense. For example, a child may make sense of the following math story as a combining story:
          "There are 6 kids playing soccer. Some more kids come and now there are 10 kids playing soccer. How many kids came to join the soccer game?"
        In other words, they see this as combining the kids who were playing with the kids who came to join them, and ending up with 10 kids. The model is 6 + ____ = 10. Even though this is additive in context, a child may solve the problem by subtracting 10 - 6 = ____. Once they've made sense of a problem, they can solve it however they want. Then go back and see if it still makes sense.

 I watched my students do this a lot the first few weeks of second grade. They would build the two parts (contextualize) then put them together and count them by 1s, 2s, 5s....however they wanted (decontextualize). They started out building everything as single blocks, and then combined the quantities into one long train. (See the picture above) They were able to go back to the context and tell me the unit, for example that the answer was a number of stickers for this particular problem. But once they made the long train, the original parts were lost to them.

Look, here was the problem: "Diva had 4 stickers. She went to the store and got another 8 stickers. How many stickers does she have now?" So the problem was that they could answer (12 stickers)...but if I then asked, "So how many stickers did she buy?" they would say "12". Because they totally forgot about the parts once the parts were swallowed up into the whole amount that was the sum.

This is problematic because keeping track of those parts is what connects addition to subtraction and it's what is going to allow us to solve subtraction problems as missing-part addition problems (for example, seeing 12 - 8 = _____ and thinking 8 + ___ = 12).

As they got better at thinking about quantities, I started making a VERY big deal about students I saw who were making the parts and color-coding them or counting by 10s and then continuing with the ones without physically moving the cubes to be together, (See picture above) thus maintaining the two parts. When I asked them I would usually say, "Which ones are the stickers she had? Which ones are the stickers she bought?" and when they could answer it was a very good thing. I'd even reserve their trays intact, to use as a mini lesson to start our next workshop....."Friends, do you see what she did??? Like all mathematicians, she arranged this so we could tell which ones were the stickers she bought! Can we answer the question of how many stickers she has now? Of course we can! AND we can still see the two numbers she made....Who remembers what this first number means? What about this second one? And how many all together? Who thinks they can work on keeping their parts like Ariel does? Off you go."

Practice Puzzles: Math Practice One

 The first few weeks of this school year, I stumbled across a way of introducing and using the Math Practices that really worked for Room 29, so I wanted to share it. I've seen lots of examples of "kid friendly" language for the SMPs (Standards for Math Practice) but this is a little more organic. What I did was, I started our Problem Solving Workshop time on the second day of school. And as my students struggled to make sense of problems, to explain themselves, to model the mathematics and find viable solutions, I just looked for ways that they were already, intuitively, using the SMPs. Little kids are natural mathematicians....never once did I have to tell them how to be mathematical. I just had to open my mind to the possibilities of what each practice might look like.....as performed by a 6 or 7 year old.

Once I had collected all eight (and I'm not going to lie, a couple were a real stretch haha) I used blank puzzles and I drew the incident right on there, labeling it all up. I am nobody's artist, but they are easily impressed and it turns out hairstyle is an easily identifiable attribute among my students. So even though any picture was only marginally akin to the child, everyone totally GOT IT.

 I gave each group of four students a tray with the puzzle pieces for one practice ready to be put together. They had a blast putting it together and then they went NUTS when they realized it was THEM. Oh my gosh! SO GOOD. :D

 

Once they had the puzzle together, they read it to each other, and made sense of what it was saying. Since they were right there when it happened, and I had made such a big deal out of each one and even repeated it over and over, they had a built-in context for making sense of each one.

Math Practice One : Make sense of a problem and persevere in solving it

            All the kids were on the carpet. I had written a simple “put together” problem on a poster at the front on my easel. “Diva had  _____ stickers. She went to the store and bought another _______ stickers. How many stickers does Diva have now?” In this process, the students have acted out the problem with a variety of numbers I supply them. She had 4 and bought 7. She had 8 and bought 3. She had 12 and bought 4. Different students act out the building of the numbers and combining them. When it comes time to do the problem on their own, I give them bigger numbers they wouldn’t really be able to do in their heads, like she had 17 and she bought 18.

            After giving them the numbers 17, and 18, I ill-advisedly did one more check for understanding. That’s when I asked Janiya what was happening in the problem. After a tense 60 seconds of silence, she slowly said, “Divaaaaa….is….she has…..stickerssssss?” Yes! And how many does she have? Janiya stares off into space. She clearly thinks I will lose interest and ask somebody else. No way, Sister. We are at an impasse, until she absentmindedly swings her head around and looks toward the poster.

            “OH MY GOSH!” I practically yell, “DID EVERYBODY SEE WHAT JANIYA JUST DID????” The other 34 students (you heard me, it was a rough first month) look at me expectantly. That’s how I imagine them, anyway. And I make a VERY big deal out of THIS THING Janiya did….because when Janiya wasn’t sure what number to build, SHE LOOKED BACK AT THE PROBLEM! Isn’t she a good mathematician? That’s what mathematicians do, when they are making sense of a problem, they LOOK BACK TO FIND WHAT NUMBERS TO USE.”

            It was a stretch, but it’s a point we’ve made over and over….’Remember what Janiya did? She did what all mathematicians do….she looked back at the problem when she needed to remember which numbers to build.”

This Week in Daily Five

I started doing Daily Five the first week of school, but it's sort of finding its groove these last....couple of weeks. No, really. It wasn't until December that I started to feel like I had my feet. It's not even that it's complicated (I'll leave it to greater minds to explain the details) but the act of figuring out how it works for me with this group of students has not been trivial. In full disclosure, I really only do it from 9:35 to 11:15, which gives me just enough time for three mini lessons and three rounds of Daily 5 3. We only do Read to Self, Read to Someone, and Word Work, though I keep committing (then backing out of) Writing. We already do a Writer's Workshop block, outside of that literacy time, so it's not like they aren't writing. It's a logistical issue for me...since there are only three rotations, 4 options means that I will have to work harder keeping track of what they've done over several days, to make sure there's balance. Friends...I'm not up to it. Just saying.

I took the plunge after Thanksgiving and had them make choices. Up until then, it was we all read to ourselves, we all read to somebody else, we all do word work. I was super worried about giving them more independence. I thought for sure some of them would not be able to read to somebody else quietly enough to also have people reading to themselves at the same time. Same for word work. And look, I'm not prone to hysteria, nor am I terribly obsessive about control. Let's just say... I had reason to believe that they would not handle this well. I also had evidence that I would not handle this well.

I was so wrong, on both counts. There is such a lovely 'buzz' of productive work during this literacy block. I love that I get to listen to every child read, individually, every week. It helps me collect ideas for lessons and small group work too. (Though the small group work is definitely my Achilles in this situation....I'm telling myself that I'm getting plenty of traction with individual conferences, but I am still hoping I'll figure this out! Tell me...how do people plan all this? and prep for all this? how many hours are in your day, I ask you??)

My friend, colleague, and former principal shared this photo from his new school, and it made me happy!

I have no control over what happens at their homes each night, but they are getting those minutes during the day, and now I want to show this to my students just to tighten up our "sense of urgency". Lately I've been feeling like they're getting a little loose with "getting started right away" and "working the whole time". I always have that feeling, actually, but when I take a step back and really look at what they are doing...I'm all smiles, I promise!

 

Read to self

 Read to somebody

(these two are like peas and carrots...love it!)

  Word Work

 I've been modeling a "Talk to the Text" comprehension strategy for the last two weeks, and they are loving it! They write on post-it notes, leaving tracks of their thinking, as they read. It can be a question, a connection, a summary, a prediction...They really love it!  

ps...these head phones serve no purpose, really. They are broken headphones that I cut the cords off of. They have no real noise blocking qualities. The kids, however, are convinced they do haha and just putting them on seems to help them focus during Read to Self.

I love finding these little "tracks of their thinking" in the books, and this is one of my favorites, found in "Pigeon Wants a Puppy". (It says, "This pegeon has lost it" haha!!)

{LOVE}

Buddy Classroom - Math

My friend and colleague is a Kindergarten teacher at my school. Although we have a ridiculous mobility rate (to wit: two students still haven't returned from their yearly trip to Mexico; one student who just came here from Egypt - Egypt! - in October is now leaving because her family found a place to live in a neighboring district; and I have a new student starting Monday from the Philippines. And that's just this week!), there is still a good chunk of my students who had her for Kindergarten the year before last. And a handful have siblings in her class right now.

A couple of months ago, we wrote them a big poster letter:

 "Dear Room 1, There are 2 ducks in the pond. Then 3 ducks came to the pond. We want to give every duck 2 peanuts.  How many peanuts do we need? We need your help! Please! Please! Please help us solve this problem! Love, Room 29

My students delivered the poster to their former teacher, and her students solved the problem by acting it out. She video taped the entire activity, from them making sense of the problem, to figuring out who should be a duck in the pond, to what should they use for "peanuts", to counting (correctly, then incorrectly, then correctly) the number of peanuts they needed to solve the problem. 

She sent the video back to me. And then I died. It is 9 minutes and 48 seconds of YES. I laughed the whole time I watched it, and I still do. It's just too good. It's hard to pick a favorite part, but watching them decide what to use as "peanuts" has got to be up there.

Teacher: Okay, so we have our ducks in the pond, what do they need now?

Them: More water! 

Teacher: Let's read the story again (reads it to them) What do they need?

Them: Peanuts!

Teacher: But we don't have any peanuts, what can we use instead?

Montrell: How about peanuts?

Teacher: But we don't have any peanuts, is there something we can use to pretend?

Elmer: We could use popcorn!

Teacher: Well, let's look at the problem, is it popcorn or peanuts?

Them: PEANUTS!!!

Teacher: Right, but we don't have any peanuts...is there something else we can use? ....maybe something in this classroom that we can pretend is a peanut?....maybe some math manipulative that could be a peanut?

Girl: How about those blocks?

Teacher: (sweating) Ok! So show us what that would look like....

Here are my students watching the video of her kids solving the problem. Along with the video, they sent a poster letter back asking us a math question too. Their question was "How do you make five?" Haha! I love them!


At first, both the Kinder teacher and I worried that it wasn't problematic enough for my 2nd graders, but it turned out it was a great question for them. We played a game of "Shake and Spill" using cups of 5 yellow/red counters. The game and the recording of how the counters landed wasn't terribly difficult for them (but it was exceedingly fun!) so our problem solving experience focused on how they could prove that they had ALL the ways, and this included making sense of the commutative property (is 2 yellows + 3 reds the same as 3 reds and 2 yellows?). They made a poster of their findings, and we sent it back to Kindergarten with a video of us playing the game and proving we have all the ways.

We are loving this way of doing Buddy Math! All of our students are working on the things that are grade level appropriate, and that are truly problematic for them, but they are still experiencing the excitement of working together through the letters and the videos.

{LOVE}

Sticky Situations

TL/DR: Scroll down and make a set of glue sponges. You're welcome.
 
People of earth: I hate glue.

I hate the gooey white glue. I have memories of spreading a thin layer of elmer's on the palm of my hand, just so I could let it dry and then pick it off. That may have happened last week. *cough* Still, I don't like to deal with it in my classroom. Either I never did it at school, or my teachers were a lot cooler than I am. I hate the way it leaks everywhere...I hate the way it clogs up and dries out in the nozzle...and I hate the way it leaves bumps under the paper they are gluing down. I am full of hate for white glue.

This problem is not solved with glue sticks. Of all the unimportant things (and there are many) that make me  feel like I'm slowly going insane in my classroom, glue sticks have to be very near the top. They don't work (papers once tacked down pop off when the glue ages), they don't last (it seems that a single child can easily plow through three sticks a month), and the kids just absolutely refuse to take care of them (I am constantly finding the lids kicked under furniture as the stick dries in the materials box). Oh, I know, I've tried so many ways to manage the glue sticks. From numbering the glue sticks, to lecturing on the importance of caring for our materials, to even guarding them with the intensity of a mother lion protecting her cubs....I can never get over the fact that they seem to be eating these things. There is no other explanation for the vortex of glue in my classroom. Sometimes, I'll walk through the room, sprinkling new sticks into the material boxes like a benevolent Fairy Godmother. Other times, I have refused to release my vice grip on any new glue sticks. I have, I regret to inform you, sternly watched 6 kids try to finish their projects before the recess bell as they pass a single, half-dried stick between them, satisfied that they are finally beginning to understand what happens when we don't take care of our 15 cents worth of glue. Get the straight jacket, I've totally lost it.

I hate glue. I hate myself because of glue. Enter the glue sponge. I found a video of it here. It's a fine video. If you are like me, you will resist watching any instructional video longer than 90 seconds. It's nothing personal, just not my preferred method of delivery. So, go watch the video. Or, check this out:

 

Pour some white glue in the bottom of a disposable sandwich tupperware.

Place a sponge (and half of another sponge if needed) on top of the glue.

Pour more glue over the top. Make sure you secure the lid correctly.

 Set it aside over night (at least).

Students take their papers, pat it firmly but gently against the sponge, and press to their project. 

No. 

Seriously. 

You only have to do this once. Just add glue as needed to refresh. (I've not had to yet, and it's been a couple of months now). The boxes have changed our lives. Two tips, one I've used and one I have not: First, keep a spray bottle of plain water; before you put the lid back on the glue box, just spritz the sponges with plain water to keep them moist. Second, if they get smelly (SUPER looking forward to this, but nonissue so far!) dribble a few drops of mouthwash onto the sponges.

{LOVE}

Spinning with The Pigeon

TL/DR: There's a pretty cute and easy free game at the bottom.

Is the weather horrible where you are? If it is, I apologize. It's almost assuredly because I have yard duty this week. When I am on yard duty, it is guaranteed that it will be terrible weather. Not complaining though....just saying. If yard duty is my least favorite thing about elementary school, that's not such a bad thing. There are so many more magical moments that more than make up for it. Sorta. ;)

One thing I love to do with The Littles is to bring things they love out from our books into the world with us. As you've heard me complain explain before, I'm doing Daily 5 with my class this year. I don't have an extensive class library to fuel the wide range of reading levels, so I've been getting beginner chapter books from the public library. I've found that if I want them to read Henry and Mudge, I have to read it to them first. Same way I got them interested in Frog and Toad. Once I read it to them, the titles fly off the shelf. One such series is the "sister" series to Henry and Mudge, called Annie and Snowball. These are stories about Henry's cousin (and neighbor) and her pet bunny.

We noticed that on every page, the bunny was sort of cleverly hanging out somewhere in the picture. And right around that same time, one of my students gave me a palm-of-my-hand sized stuffed white bunny. So I started hiding the bunny around the room, and when they'd discover it, I'd move it when they went to recess or left for home for the day. Then they'd discover it again, over and over. Oh my! So much joy in that moment! I just laugh and laugh when they point and jump and exclaim over where they've found that silly little bunny!

Here's the little bunny on top of the loudspeaker in our classroom. 

Today, they found her perched on top of the overhead screen.

Okay, back to math! haha! This is actually a mathy story, because for our math games, which I'm in the middle of rotating out, I made a fun little spinner game (for recording: spin, tally, graph.... add/compare options) and they can't get enough of these activities. They just think they are so fun! So for this one, the four options on the spinner are each from a book we've read together. The bear from Ira Sleeps Over, the purse from Lily's Purple Plastic Purse, that crazy bike from the original Splat the Cat, and finally a hot dog from The Pigeon Wants a Hot Dog. They loved these books, and they loved seeing these old friends incorporated into a math game. 

You can download from my google drive here:

* Spin and Record with Literature Connections
* Y por los chicos y las chicas

Logistics: These are color prints. For the math box, I print 3 copies of the spinner itself (which go into sheet protectors), throw in a few generic clear spinners that they overlay on the spinner page, and include a matching number of recording sheets, also in sheet protectors. In the tub are also 3 expo markers and a swatch of felt they can use to erase their work on the plastic protector of the recording sheet. I don't have unlimited ink budgets, so this is a good compromise. I also reserve the right to print the color pages in black and white, but they seem fine so far with just erasing their work at the end of their work period.

Matching Cards

I made these matching cards right around the time school started, and I kinda love them. If you love them too, you can download the pdf files from my google drive. I use all the sets with my second graders. I just photocopied each set onto a different color of cardstock, and they keep the whole stack in their shared materials buckets in their table groups.

If they finish their journal and calendar work before their classmates, they can pull out the cards and play "concentration" style to match the cards.

 They can play alone...

Or with a partner or small group.

They love these cards as much as I do! They are always available, and I have to admit, they have saved my bacon more than once completely outside of our math block. Nothing like a no-prep no-fuss activity that can be used anytime they have a few minutes to spare.

Me: Why are you running with a broom?

Them: I'm done!

Me: Put that down and get out your math cards. Please.

*ahem*

Because we want our students to use the standards for math practices, it's been equally beneficial to have some more structured time with the matching cards, too. For example, I'll have them all use the same purple set, and they need to take turns putting two cards together with an explanation for why they are choosing a particular pairing. Then everybody in their group has to agree (thumbs up) or disagree (offer a different pairing and explanation).

Click here to download:

* Numbers 1 to 9, match to pictures of one to nine objects

* Teen numbers 11 to 19, match to double ten frames (10 + extras)

* Random Numbers to 20, match to tally mark representations

* Ten frame representations of 9 +, match to ten frame reps of 10 +, match to numerals 10 to 18

* Ten frame representations of 9 +, match to numerals 11 to 19, match to "9 + ___ is 10 + ____"

* Numerals between 10 and 150, with base-ten "ten stick" picture representation