Reengagement Lesson: Modeling our Problem

It's the same every year. I think my sanity is in jeopardy every time I start problem solving the first week of second grade.....and the struggle is real. But within what amounts to just an hour or two in real time, they are making great progress towards getting sensible answers, showing their work on their trays, and being able to talk to their peers about what they are doing and why they are doing it. 

After three sessions of working on our Problem of the Month, I was elegantly reminded that all little kids really need are some good questions, tools, and space. As hectic as it feels, they are actually running it like a boss. Now it was time to get some paper and explain how they got their answers.

Oh boy.

About two-thirds of it was unusable. Of the remaining one-third, I had to really think critically about what my realistic goal was for this first time out. I had a handful of students who wrote out what they did in words; another handful that drew pictures of what they did; and another few who wrote a number sentence AND drew a picture. Of those few, two of them were done correctly, and here's a picture of one of them:

I think this one is so interesting for several reasons. First, the notation is really great. Most students would be thinking about taking away 8 and 5 acorns, which matches the problem situation. But this Little added them...to each other....and then subtracted that subtotal from the 17 acorns Cevanna started with. (If you clicked on the link above, you know that Austin is the child in the problem, but I usually change names to be my students.) As much as I love this one, I had to pass. I needed a straighter path to my teaching point, which I had decided would be:

Mathematicians draw pictures to match their problems, 

and then they write number sentences that match their pictures. 

They do this by thinking about how they started, 

what changed, and how they ended up.  

Writing a teaching point out in this format (Mathematicians do this _______ by doing this______) has been so helpful for keeping me focused on my learning outcomes. I can't recommend it enough. It's the best way I have found to keep me dialed in on the thing I want my kids to take away from this problem to use with their next problems. As you can see, the above recording is clever and thought provoking, but the "what changed" part involves some extra hoop jumping that two-thirds of my class was perhaps not ready to ponder. They will get there, but not if I skip ahead to a more advanced number sentence.

I decided to go with Madai's explanation, because most of my students were relying heavily on pictures.  I wanted to start with a recording that they were already trying to use. Again, most were not very productive drawings, but she is killing it here, and I thank her.

 ("There are 17 acorns. Cevanna gave 8 squirrels an acorn. Then she gave 5 

more acorns away. Now she can feed 4 more squirrels.")

I started this lesson by telling them that mathematicians draw their pictures to match what's happening in the problem. I posed the question, "How do we know Madai is matching her picture to the problem?" With some turn and talk, help from Madai, and just a wee bit of wait time, we were able to articulate that Madai made 17 boxes to be Cevanna's acorn, and then we circled the 8 she crossed out to show she had given them away, then we circled the 5 she had crossed out to show she had given them away. Then we counted what was left. See how focused we are, thanks to our teaching point?  "Mathematicians draw pictures to match what is happening in the problem."

 Next, I showed Ethan's recording. (pictured above) We had some more conversation just trying to interpret what he was doing with his drawing. They asked him if he was showing the acorns and the squirrels. He confirmed he was and then used a pointer to show how his drawing matched the problem. I posed the question, "Madai used crossing out to show what she took away....Ethan hasn't crossed out anything. But he is showing how Cevanna is giving away her acorns.  How do we know what he is giving away, if he isn't crossing any boxes off?" Very fruitful discussion. {LOVE} At the end of the discussion, we repeated that "Mathematicians draw pictures to match their problems, and then they write number sentences that match their pictures" and they agree that it's just exactly what Ethan did.

Finally, I showed them Justine's work, pictured above. (Justine, pronounced Justin #awkwardrollcallfirstdayofschool) Here we noted that he used neither numbers nor crossing off to show what he had done, but we spend a moment counting out and interpreting his groups. I had photocopied (shrunk onto half pages) his work and I gave each child a copy of this picture with the reminder that "Mathematicians draw pictures to match their problems, and then they write number sentences that match their pictures. They do this by thinking about how they started, what changed, and how they ended up."  

I invite them to try to do just that, the way mathematicians do. I used these little sheets as their "tickets out the door" and I was pleased to see that 17 students were able to write the number sentence 17-8-5=4 to match Justine's picture. That means 11 students were not able to write the matching number sentence. I don't plan to address this again in this context, but we now have this anchoring experience that I can refer back to as we move forward trying to record drawings and number sentences for our future problems.

Here are the things we learned from this problem, that we can use in the other problems we do:

1. The most convincing explanations show drawings, number sentences, and words.

2. Your drawing should match what is happening in the story and 

your number sentence should match the drawing and the story.

3. When we write number sentences, we think about how we started, how we changed, 

and how we ended.

(I made this list into an anchor poster, attaching the student recordings we had used to surface these main points. We will refer to it often over the next many weeks and months, and we will add on to it where we need to as we clarify what makes a powerful explanation.)

How to Keep a BIG PROBLEM Going

Doing BIG problems with LITTLE kids involves a really beautiful set of challenges. Beautiful, I say, because though the obstacles are great, The Little Ones are capable of getting on board in endlessly satisfying ways.

One of the biggest challenges we face is trying to keep them engaged and pushing their thinking forward over a series of work sessions. Inside of that challenge is the fact that there are so many ways to be wrong, it can be hard to pick the most important issues to tackle or to pick the trajectory of skills and behaviors that will lead to the greatest growth in content. You've got attitude AND aptitude that you have to address, and it can feel a little bit like putting clothes on cats....maximum hustle in random directions while somebody is trying to scratch your eyes out. Too dramatic? 

I wanted to share a few things I think about when I'm planning for future lessons to push us on a Big Problem. First of all, I consider every work session to be an adventure in formative assessment. I'm constantly looking for strategies or behaviors that I want to promote. It's helpful here if you let go of "the right answer". There's time for getting right answers. I always say I'm not the kind of hippie who says "It doesn't matter if you're wrong, as long as you can explain yourself!" Actually, it does matter. It's math. There are answers in math, and I want my students to be good at getting them. However, having said that, if all I'm focusing on is that right answer, I'm missing opportunities to build their capacity to be mathematical in really challenging situations. I'm diminishing their long term ability to get right answers in powerful ways if I stop every half-baked thought or idea because it's not RIGHT. When I'm teaching the way I aspire to teach, we have to identify, exam, and discard a whole lot of "wrong" before I can say they totally own "right". That takes time, and letting go of nitpicking their answers can create space for the bigger work we are trying to do.

Second of all, I think about the Standards for Mathematical Practices (SMPs). There are 8, and they all focus on the behaviors of mathematically proficient students. If I know that I want to see students doing these things, I look for students who are doing them, and we name them and invite their classmates to do them. 

Finally, I look at misconceptions my students are nursing along. Misconceptions can manifest in very sneaky ways, and it's important to get practiced at ferreting them out...a misconception that isn't addressed will be cleaved to in really unattractive ways.

SO:  

1. Strategies I want to promote

2. SMPs students are doing

3. Misconceptions to address

Each Problem Solving Workshop follows the same structure....a mini lesson, a work session, and a closing lesson.  I wrote about introducing the problem here, Part 1. The next day, I did a mini lesson on what the word "each" means, here Part 2.  That same day, but written about separately here Part 3, I set the kids to working. I chose the following trays from that work session to share with the kids as a mini lesson to start the next work session the following day.

Focus of our Mini Lesson:

SMPs students are doing -SMP #1 

This tray just shows that a student started by making a train of 17 cubes to match the problem, which I always have posted.

This is what SMP #1 (make sense and persevere in solving problems) looks like with The Littles. During that first work session, some students couldn't really get started. They fumbled around a bit. Good job, Littles! You stuck with it! But we aren't just cheerleaders, we need to help them develop real strategies for getting on it. I showed them the tray and said, "How does this tray match the problem? Why did this person do this?" After some private think time, a little bit of partner sharing, and some questions for clarity, they told me that this makes sense because the problem says 17 acorns and these are 17 cubes, which are the acorns from the problem. I pressed on them until they articulated that if you weren't sure what to do, you could go back and look at the problem and see how it starts, and try to make your tray match the way the story starts.

And here we go! During the work session that day, they quickly got busy building their 17 acorns. Are you surprised that we are doing the exact same problem we already did? Pro tip:  don't introduce new problems every day. I like to share with them my favorite George Polya quote, "A problem isn't a problem if you can solve it in 24 hours". We are doing this problem again because that's what mathematicians do, they stick with their problem until they have learned everything there is to learn about it and from it. It's a good thing!

Strategies I want to promote

I followed the same process for the next two trays, just placing one in the middle of the learning circle and giving them some prompts to think about and partner talk about. Mostly, I just asked "What did this person do? How do you know it matches what is happening in the problem?"

And then this one, with the same prompts:

This one is different because it shows the squirrels. In Level B of this Problem of the Month, there are different kinds of squirrels and each type of squirrel can carry a different number of acorns back to their homes. Students will need to figure out how many trips the different types of squirrels will take to get the acorns back to their homes. In any case, managing the tools to represent both acorns and squirrels, in addition to the clever color coding to keep track of the different groups of squirrels, is a really helpful and powerful strategy for doing Level B, which we will be doing soon, so I wanted to make sure we put it out into our universe so we can call upon it.

And during the work session, after this quick inquiry mini lesson on these two different strategies for keeping track of feeding the squirrels in our problem, I saw the students experimenting.

 Various stages of making sense..... This little group had a hard time finding their error (they started with 16 acorns instead of 17 and thus ended with 3 acorns instead of 4) because they all made the same mistake and easily convinced each other that they were killing it. I made the executive decision to not address this wrong answer in real time, and instead held space for them to practice the strategy and practice their recordings.

 Another error....since this child was working alone and hadn't worked himself into a lather about how right he was, I found it easier to do the "match making"... This student was paired up with another student who used the same strategy, but who made 8 squirrels and 5 squirrels instead of 5 and 5. The dream is that when they talked it out, the other child would be convincing enough that this child will see how to fix their error without me being involved at all.

 I love this one, where he uses the tiles to be squirrels instead of the counters. I think little details like this show that he is really making sense of the strategy and making it his own, as opposed to just "mimicking" the strategy of using counters but not really understanding what he is representing. Kids have that kind of magical thinking sometimes, as though the math is in the chip instead of in the chip as representation. Recognizing that he can use any tool to show his squirrels is a little thing that feels very big in a primary classroom.

You nailed it, kid.

Now go find somebody to talk to!

Misconceptions to address

I saw wrong answers, but no real misconceptions. You can do everything "right" and still make mistakes. In life and in math, it turns out.

That was it for this session! Imperfectly perfect, in my eyes. So much movement in just this one attempt, and so many places still to go. That's how you know it's a good problem :) My "closing lesson" was on how we put away our trays and bags. Not life changing in terms of content, but lookit, these conversations have to happen. For my sanity.

First Problem of the Month - PART THREE

So we started this POM with the Primary Version on Thursday, and continued with introducing Level A on Friday. We did our first work session on that same day, but I'm writing about it separately because, in my mind, it was a whole other animal. For those directed explorations and that introduction, I was very much at the center of those activities. (Evidenced by the number of times I say "I" in the last post *ahem*)  I'm not saying I don't engage my students, because I do, with questions and protocols and prompts for getting them talking and thinking. But it's just not the same as saying, hey, here's this situation....what would you do with it? The noise level (theirs) and aggravation level (mine) are proof that these are not the same lessons.

We read the problem together and I dismissed them to grab their math bags and a tray.  "You get what you get, and you don't throw a fit" when it comes to the trays people, let's keep moving. Roughly half the class got back to their desks and immediately started in with "What are we doing? I don't know what to do." Sigh.

I started to coach in with a kid or two, asking them what's happening in the story, can you make her acorns, but I gently reminded myself that that's not my job and instead, I ignored the escalating noise of the frustrated masses(save the occasional warning that "We do not swing our math bags!" Safety first, people, safety first)  and started trying to focus in on the students who were making sense (there were a few!) and what their strategies seemed to be.

After about five minutes (which felt like 4 months), I called for a "mid workshop interruption" and told them we would be putting our hands behind our backs because you're not allowed to touch anybody's work, and we would be walking quietly to look at other people's work to see if we could find any ideas that might help us get started back at our own trays.

Here is a picture of that. There are 20 boys in this class of 32.  Can we get a moment of silence for their teacher, please. (But don't feel TOO badly for her...she is charmed.)  In the interest of full disclosure, I had to put one child on a time out right after this for destroying a child's work, just until I could get over there and have a little heart-to-heart about what "we don't touch anybody else's work" actually means. I also had another student who decided to loudly announce, as he stood over each tray, either "LOVE IT" or "HATE IT". Which started a chorus of this in the class and required me to call for a moratorium on all proclamations.... just look people, and zip it for now. That's some solid teaching, right there.

But even with these nuisances, it was well worth it when we got settled back into our desks and we reached my magic number of around 75% of my students working hard on whatever their idea was.

Here is a thirty second video of me verifying that this Little One can articulate his first step.  We stand up and within seconds we are at a desk where his services are sorely needed. After he gets her started in counting her 17 acorns, I pull him to work with my other friend here on the carpet.

Both students started by making 17 acorns, but Andrew (on top) interpreted giving "each one" an acorn as giving away one acorn, whereas Junior, below, is correctly interpreting the problem. I love this kind of "match making" and use it on the daily. My only prompt was, "You two come down to the carpet, I want you to listen to each other, and see if you can understand how the other person solved the problem."

This is another student who used Andrew's idea (they were working together) but he is using his tools differently. This is so common, I see it every year, where students use these base-ten sticks as tallies or some other kind of unit.

I have no idea what they were talking about, but I love the spontaneous crowd that gathered around as she walked them through her thinking.

I like how a large number of students used these sort of "grouping" strategies to keep track of the different "parts" of the problem, being the 8 acorns Cevanna gave away first, the 5 she gave away next, and the four remaining in the little stack in the corner.

I also love little Rodrigo's use of color coding.  The red ones are the 8 squirrels and the yellow ones are the 5 squirrels, and the cubes are the acorns he's passing out.  Nice matching of the cube-to-counter also. And clearly, his way makes a lot of sense to Princess, next to him, who is co-opting it as her own. He was really good at explaining himself here, too. {LOVE}

It was bumpy, I'm not going to lie about that. But in the end, I was really pleased with their overall progress and attitudes. I know we will do this exact same prompt on Tuesday, when we get back to school after the three-day weekend (bless) so I wasn't worried in the least about who got what answers. My biggest concerns were around the culture I wanted to build in math class. By far, this class was the most aggressively negative in the way they generally dealt with each other (mathematically speaking) and it made me sad. But no way was I letting them see that! So I passed on a lecture and closed by making this quick poster:

I posed it as simply, "this is how mathematicians talk" and we found some words to replace the negative ones I'd been hearing during the workshop. My favorite is "Who wants to talk about the problem?" instead of "I'm done" because nothing goes straight up a teacher's back more quickly than a kid who yells that they're done 90 seconds into a carefully crafted lesson. Really now. Oh, and as for that final one... They made the cutest connection to this book which our Instructional Coach had come in and read to them the day before. When we replaced "I don't know what to do" with "Can somebody help me get started" they said it was just like the story, you just needed an idea that you could feed and grow. {LOVE}

I think I'll close by reminding myself that going slow helps you go fast. And then I'm going to remind myself that, really, they were pretty amazing. Then I'm going to take a nap.

First Problem of the Month - PART THREE

So we started this POM with the Primary Version on Thursday, and continued with introducing Level A on Friday. We did our first work session on that same day, but I'm writing about it separately because, in my mind, it was a whole other animal. For those directed explorations and that introduction, I was very much at the center of those activities. (Evidenced by the number of times I say "I" in the last post *ahem*)  I'm not saying I don't engage my students, because I do, with questions and protocols and prompts for getting them talking and thinking. But it's just not the same as saying, hey, here's this situation....what would you do with it? The noise level (theirs) and aggravation level (mine) are proof that these are not the same lessons.

We read the problem together and I dismissed them to grab their math bags and a tray.  "You get what you get, and you don't throw a fit" when it comes to the trays people, let's keep moving. Roughly half the class got back to their desks and immediately started in with "What are we doing? I don't know what to do." Sigh.

I started to coach in with a kid or two, asking them what's happening in the story, can you make her acorns, but I gently reminded myself that that's not my job and instead, I ignored the escalating noise (save the occasional warning that "We do not swing our math bags!" Safety first, people, safety first) of the frustrated masses and started trying to focus in on the students who were making sense (there were a few!) and what their strategies seemed to be.

After about five minutes (which felt like 4 months), I called for a "mid workshop interruption" and told them we would be putting our hands behind our backs because you're not allowed to touch anybody's work, and we would be walking quietly to look at other people's work to see if we could find any ideas that might help us get started back at our own trays.

Here is a picture of that. There are 20 boys in this class of 32.  Can we get a moment of silence for their teacher, please. (But don't feel TOO badly for her...she is charmed.)  In the interest of full disclosure, I had to put one child on a time out right after this for destroying a child's work, just until I could get over there and have a little heart-to-heart about what "we don't touch anybody else's work" actually means. I also had another student who decided to loudly announce, as he stood over each tray, either "LOVE IT" or "HATE IT". Which started a chorus of this in the class and required me to call for a moratorium on all proclamations.... just look people, and zip it for now. That's some solid teaching, right there.

But even with these nuisances, it was well worth it when we got settled back into our desks and we reached my magic number of around 75% of my students working hard on whatever their idea was.

Here is a thirty second video of me verifying that this Little One can articulate his first step.  We stand up and within seconds we are at a desk where his services are sorely needed. After he gets her started in counting her 17 acorns, I pull him to work with my other friend here on the carpet.

Both students started by making 17 acorns, but Andrew (on top) interpreted giving "each one" an acorn as giving away one acorn, whereas Junior, below, is correctly interpreting the problem. I love this kind of "match making" and use it on the daily. My only prompt was, "You two come down to the carpet, I want you to listen to each other, and see if you can understand how the other person solved the problem."

This is another student who used Andrew's idea (they were working together) but he is using his tools differently. This is so common, I see it every year, where students use these base-ten sticks as tallies or some other kind of unit.

I have no idea what they were talking about, but I love the spontaneous crowd that gathered around as she walked them through her thinking.

I like how a large number of students used these sort of "grouping" strategies to keep track of the different "parts" of the problem, being the 8 acorns Cevanna gave away first, the 5 she gave away next, and the four remaining in the little stack in the corner.

I also love little Rodrigo's use of color coding.  The red ones are the 8 squirrels and the yellow ones are the 5 squirrels, and the cubes are the acorns he's passing out.  Nice matching of the cube-to-counter also. And clearly, his way makes a lot of sense to Princess, next to him, who is co-opting it as her own. He was really good at explaining himself here, too. {LOVE}

It was bumpy, I'm not going to lie about that. But in the end, I was really pleased with their overall progress and attitudes. I know we will do this exact same prompt on Tuesday, when we get back to school after the three-day weekend (bless) so I wasn't worried in the least about who got what answers. My biggest concerns were around the culture I wanted to build in math class. By far, this class was the most aggressively negative in the way they generally dealt with each other (mathematically speaking) and it made me sad. But no way was I letting them see that! So I passed on a lecture and closed by making this quick poster:

I posed it as simply, "this is how mathematicians talk" and we found some words to replace the negative ones I'd been hearing during the workshop. My favorite is "Who wants to talk about the problem?" instead of "I'm done" because nothing goes straight up a teacher's back more quickly than a kid who yells that they're done 90 seconds into a carefully crafted lesson. Really now. Oh, and as for that final one... They made the cutest connection to this book which our Instructional Coach had come in and read to them the day before. When we replaced "I don't know what to do" with "Can somebody help me get started" they said it was just like the story, you just needed an idea that you could feed and grow. {LOVE}

I think I'll close by reminding myself that going slow helps you go fast. And then I'm going to remind myself that, really, they were pretty amazing. Then I'm going to take a nap.

First Problem of the Month of the Year - PART TWO

On Thursday, I started the POM "Squirreling It Away" with my new second grade class. (Part One is here.) Friday was Day 7 of the school year, so why not open up a can of crazy? Level A reads that there are 17 acorns, and we are going to feed 8 squirrels an acorn each, then we are going to feed 5 more squirrels an acorn each, then we are going to answer the question, "How many squirrels can we still feed?" If you think this is an easy problem, it is because you are  a) not seven years old and/or b) not dealing with 32 seven-year-olds. The logistics of them getting out their math tool bags, passing out trays for them to work in, and getting them to choose which tools to use for this problem takes about 3 years off my life.

I think a lot about how to introduce a problem without discussing how to solve it. I think here I decided on a two-pronged approach, really. First, I knew we were going to read it together, and I knew it was akin to what they had acted out the day before (though, I will give you a dollar if even four of my students made that connection :-/), so I wanted to simply unpack the word "each". I predicted that if they could understand the mathematical implications to the word each, these Littles (with no formal instruction in multiplication) would have a good jump on how to attack the problem.

Second, I knew I did not want to tell them what "each" means. Instead, I used my favorite unpacking- a-new-idea device... I gave them a non-example. Even very young children have a lot of intuition about how math works. Tapping into that intuition is as easy as giving them a counter example and letting them talk about why it doesn't match the problem situation. This keeps the cognitive demand I'm placing on them much higher than were I to simply demonstrate what "giving one to each" squirrel means.

So I whipped up a quick little powerpoint that I could project to broach the subject. The first slide, I replaced "Austin" from the problem with "Cevanna", one of my students, and I put her picture in. I don't know why, maybe because it's my habit to sort of math-harass all my students, but I didn't mention there were 17 acorns. I just put them in and made them tell me. After giving them a moment to count, it turned out that we weren't in agreement and two or three different volunteers got to point with the pointer while we all counted them out. Yes, there are 17 acorns in the bag.

(Sorry about Cevanna's creepy face blurring, I just can't remember if I have permission to share her image and the paperwork is in my classroom, so this is what we get. Also, the "17" on this slide is animated, so after we counted and recounted and convinced ourselves, the 17 comes in.)

This slide is also animated. The 8 squirrels are there, as is Cevanna, but we read the prompt on top and then an acorn flies in and attaches to the squirrel. When I ask if this is how she fed the squirrels they are all "yes" and they are all "no" in equal parts. The bottom prompt comes in and without further ado they turn and talk.

(Here is a copy of the powerpoint that can be downloaded and edited to add a student's picture, if you want to try it this way with your own class.)

When I call them back, and I ask if this is what Cevanna did, they unanimously tell me NO. I call on a few people to describe why it's wrong and after several tense minutes we have a really viable argument about how this is not a model for each. I'm the only tense one, by the way, they are having the time of their lives.

After this, most of the class is good to go! I poster up the problem and we get our bags and trays and it's on.

First Problem of the Month for the Year - PART ONE

I have never kicked off Problem Solving Workshop with a Problem of the Month before in my second grade, but I believe in mixing things up and so that's exactly what I did on Thursday.  We did the Primary Version of the POM "Squirreling It Away" first. The Primary Version is one of my favorite ways to start a POM, since it is very often a scaled down version of the Level A and it helps with giving kids access to that first level.

We acted out having 5 acorns, and one squirrel coming. We gave the squirrel an acorn. How many acorns do we have left? We repeated this for 6 acorns and 3 squirrels; 4 acorns and 3 squirrels; 7 acorns and 2 squirrels.

We then retreated to our desks....or whatever various places I'm currently stashing 32 kiddos. From here we did the two-part problem of the Primary Version, where we have 10 acorns and 5 squirrels come up and we give them acorns and then 3 more come and we give them acorns, and the question is then "how many more squirrels can we feed?"  We did this for several combinations (12 acorns to 3 squirrels and then to 4 more squirrels; 9 acorns to 2 squirrels then to 4 squirrels).

I had to remind myself several times that this is what it's like at the beginning of the year. During these early attempts, I can find myself leaning towards answer-getting, towards just getting through the activity... students often approach problem solving work with an intellectual laziness that is frustrating to work around. In order to create the kind of math environment that I want, I stick to my 75% rule...if 24-26 of my kids are paying attention and direct modeling the problems, I just ignore those who are not. I'll get them eventually, I just have to forgive myself if I don't get them ALL the FIRST time out....otherwise I get very cranky. Just saying. If less than three-quarters of my kids are engaged and moving along, I chalk it up to "that's the wrong problem" and we just clean up, no harm no foul, and we try again tomorrow. In this case, really only a handful of students were giving me the blues, so after a quick assessment of who's doing what, I put on my blinders and we soldiered on.

We finished this part on Thursday and I brought them back to the carpet for a quick lesson on what it means to record what we just did with our blocks. We did it with drawings and number sentences, but it was necessarily imperfect since I was relying almost totally on what they could supply.  If I asked, "How could we show that we fed 5 squirrels with these acorns" and they answered, I had to use it. It's the first pass of even opening up the discussion that "this math that we are doing, it looks like something when we write it down" so I'm not worried. We spend all year developing this fully.

Then I went home and took a nap. The end.   :)