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 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.

Second Problem Solving Workshop

Last Thursday was the first session (described here), and last Friday (the 8th day of school) we did the same problem with new numbers.  After doing 22-7, I noticed that most kids could get an answer, but there was very little understanding of place value (as tens and ones) and regrouping (in order to get more ones from a ten). Totally normal in the beginning of second grade, from my experience.

The workshop period follows the same structure of writing or reading workshop:  a mini lesson (10-15 minutes), a work period (115-20 minutes), and a close (5-7 minutes).  In this case, I wanted to address the way we build "bigger numbers", tool choices, and then sharing one strategy that a student had for regrouping. Then I would send them off to work on the exact same problem with even more complex numbers.

On the first day, I had seen many ways to make 22, but I didn't want to encourage all of them. There were ones I didn't love....here's one:

He was using the ten sticks like units, or tally marks, and needed 22 of them to make 22. It's not unusual, and he will figure this out (as of today, the 11th day of school, he was still doing this). I'm not sweating this (yet, haha) but I'm not offering it to other students, either.

So here's my mini lesson, to open up day two of our Problem Solving Workshop. I drew out the first tall train of 22 cubes, and the second 2 ten sticks with two extras way.  The last one, with two ten-cube trains and two extras, was added during the lesson, as were the marks cutting across the first tall train way.

I began by saying I saw lots of ways that people made 22...I said, I saw Mehdi make this tall train. (Added his name.)  When I asked if anybody else made it this way, we got lots of "me too" hand motions. I told them we would have to count them, to test his train, which we did chorally. When we verified that it was 22, I wrote it under the train. 

I went on to say that I saw some people making it this way, like Angel did. (Added his name.)  This also got lots of "me too" hand motions. (For "me too" we just do a thumb pointing to our chest, pinky stuck out in front of us, almost like a "hang ten" but pointing - often frantically - back at ourselves.) We counted it as 10-20-21-22, labeling as we went along and writing it under the train once we had verified it.

Now, my goal, always, is to get them talking and listening to each other. But it's with a nod to my sanity that I do a bit more of the heavy lifting in this area at the beginning of second grade than I am totally comfortable with. I can't tell you how many times I've started a lesson with "Jasmine did the most interesting thing yesterday, Jasmine, go ahead....tell us what you did" as a way of jogging her memory and handing off the discussion to a student, only to have the student launch into a totally unrelated, inconsequential account that usually starts with something like, "oh, first, I took all my blue cubes" (no you didn't) "and then I put 2 and 5 and 3 and then I..." (no you didn't) "I thought about what I should do and then I remembered that I had some red cubes" (no you didn't)..... enough already, let me handle this.

So Jasmine had built her 22 like this:

She had originally built it with the two ten sticks and the two extra cubes, but when I came back, she had built the second way, with the two trains of ten cubes and the two extra cubes.  When I asked her about this, she showed me how she took away the two, but then couldn't break the sticks:

Haha here she is trying to snap off a couple. So cute. During the mini lesson, I asked her to show us how she tried to snap some off, and at this point, I was able to say, can you now show us the NEW way you made the number 22? And she was good to know, she knew exactly what I was talking about and there we went.

So far, in the 10 minute lesson, we had addressed building the quantity three different ways, and we had shared a strategy for regrouping by using the cube trains rather than the base 10 ten sticks to build the number. (Notice that she didn't just regroup one ten, she redid ALL the tens. In direct instruction, we would no doubt instruct her to regroup one ten...virtually every second and third grader I've ever met does it the same way as Jasmine, before they make sense of just swapping out one.)

To close the mini lesson, we went back to Mehdi long train and I asked them to partner talk about if Mehdi's way was the same or different as Jasmine's way.  After we discussed it, I posed the question:  Can we make Mehdi's look like Jasmine's? Are there ten-trains inside this long train? (The said yes, there are)  How many do you think we can get? (two) Let's try. (We counted up to ten, marked it off, counted up to ten, marked off....and saw the two extras, just like Jasmine's. I invited them to think and build their numbers using "TENS" and gave them their new numbers.

We did the same problem and I gave them the numbers on the yellow post it note (34, 18). We read the problem together as "Ishika has 34 shells. She gives Jaiyana 18 shells. How many shells does she have left?"

 

And here's Mehdi....he moved immediately from the long train to the ten-cube trains to make 34.  Proving, once again, that sometimes a kid hears the exact next idea they needed to hear, and that's all they will need to move forward.

 And here's Bryan and Ahmillyion making tall towers. Proving, once again, that when a kid is not ready to hear it, they will take a great idea....and do absolutely nothing with it.  No big deal, I will be inviting them to think in tens for the next few months.  They will get there!

During this work session, Angel continued to build his numbers with base 10 ten sticks and single unifix cubes.  When I asked about how he was giving away 18 shells, he showed me how he gave away as many as he could, then he used his finger to count down the markings on the permanent ten stick. He held his finger over the counted off section and said, "If I could take this off, I would." I told him about Brandon, a third grade student I had many years ago, who had this same idea and he discovered that he could mark them off with an expo marker. We got one, and I showed him how it would be fine, that it rubs right off, and he went right to work.

Here he has crossed out all of one ten with a straight line, then x-ed out 8 more from the other permanent sticks. He counted the remaining cubes as 1-2-3-4-5-6 and 10 more is 16.

Lovely!  Now we have successfully concluded our SECOND problem solving workshop. Students are making sense of a simple give away problem in context, and we now have several ideas for building bigger numbers with tens, and two ideas for regrouping (marking off, or swapping out) when we don't have enough ones to give away.  I will point out, that there is no way I could give my second graders a worksheet with problems like 34-18 or even 22-7 during the first week of school. It is only because they are using this "direct model" method that they are able to do the math, while simultaneously making sense of place value, and developing their understanding of the attributes and functionality of the different tools.

The close on this second day was straightforward:  I saved Angel's idea for the opening of the next session, and we cleaned up and reconvened on the carpet to recollect what a responsible classroom sounds like and looks like when it is time to clean up. Hint: there is no yelling, running, or swinging math bags over our heads. *ahem*

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}

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

In Math...Using Cognitively Guided Instruction (CGI) for Problem Solving

 We read our first problem together, with no numbers. Jose and Khazjon are students in our class, and I picked Lego Ninjagos as our math currency because it didn't take me long to figure out where our interests lie. Once we read the story, I asked them what was happening in the story in their own words. "Jose has legos!" Mmhmmm....do we know how many Legos? "No! I think he has 5." hahahahah so funny this always happens! I can't tell you how many times I write a problem just like this, then when we finish reading "How many Lego Ninjagos does he have left?" I turn to find a forest of little hands, waving in the air, wanting to "answer" the question. LOVE

So what else do we know about our story? "He gives Khazjon some of his Legos." Do we know how many he gives him? "No! But maybe he gives him 10, I think." Maybe! What else do we know. "Jose has some left." How do you know that? "Because we have to count them." And what will that tell us? "How many he still has." Who is he? "....." It says "he" still has....who is "he" in this story. "...Jose!" Yes, we are finding out how many Jose still has. Our school is two-thirds English Language Learners, and I have learned that these referent pronouns are tricky...it is not always obvious who "he" refers to...is it Jose? Or Khazjon? They can figure this out, but asking the question surfaces it for them, and they sort it out, before we get into problem solving.

Next, I put two numbers on the chart, under the story, and we reread the story putting the numbers in order as they come up. We did 8 and 3, then we did 12 and 5, then we did 14 and 6. I have a small tub with a variety of tools, similar to their Math Bags, and two students (first Jose and Khazjon, then students acting as Jose and Khazjon...boys and girls alike....good times, good times!) act out counting out the starting number of Legos, then act out giving some to the other person, then we predict what is left in their hand, and I always ask, "How can we prove it?" I get either "because 3 and 5 is 8!" or else "we can count them!" and we always do. "So we think there are 5 in Jose's hand right now, let's count and see if we are right!"

Once we understand the problem, I set them off on their own with two final numbers. Because I'm a masochist, I gave them the numbers 22 and 7.

 They did a spectacular job! They grabbed their Math Bags and a tray, and with the vague instructions to "show me what the problem looks like", they got right to work. Here we see Dontrell and Jossah using two different tools to build the 22 units, and then they broke off or removed 7 tiles/cubes and counted what was left. Nice solid CGI strategy.

 

Now, I have worked with third graders who lost their collective minds when they built their numbers with base ten blocks and didn't have enough units to give them away. I've seen some seriously crazy stuff. But this Little One (second grade) took it in stride. She built her 22, pondered it for a few minutes, and quickly changed paths to make 22 unit blocks so she could give away 7. Please to note, she did not "exchange" one of then tens for the units, which would have sufficed, but instead completely rebuilt the number using all unit blocks. I see that efficiency is not on her mind! Good job, Abbs, you knock my socks off!