Dot Talks - The good, the bad, the ugly

I spent all summer telling teachers to start their Number Talks with dots. I mean, I always do, and I have video and photographic proof that it works. My third graders came in 2 to 3 years below grade level (and I'd always joke, "How are they 3 years below grade level? They are in THIRD grade! Did NOTHING happen in three years?" but really...it's not funny) and I swear by dot talks to build their number sense and make things happen from the first day of school.

As a math coach, I did "dot talks" in hundreds of K - 6th grade classrooms. They are a miracle, I swear it!

So it is with great humility that I relate that, on the fourth day of doing dot talks with my new second grade class, I FINALLY did one that worked with this group. Oh. Em. Gee. The first three weren't just bad....they suuuuuuuucked. Big Time.  And it hurts because I think I know where I went wrong.

I'll start at the beginning. I used this dot talk on the first day of school, and it should have worked. I've done it on the first day of school for 1st, 2nd, and 3rd grades.

The general protocol is to show the card with the dot configuration and ask "How many dots do you see?" I show the whole class, then drop the card. (If you hold the card up the whole time, they will just count them, one by one. I want them to rely on parts that they see to put together the total. Counting still happens, but I want them counting the parts they see, not just the whole thing at once.)

When I asked, "How many dots are there?" I got from 4 to 11. Hmmm....should have been my first clue?

When I look at our recording poster, I have to admit, it doesn't look that bad. But it was painful, my friend, I cannot lie. Here, see what I mean?

 Still, no worries, I have had terrible dot talks. I mean, as much as I love them and believe in them, there is no question that they can go very badly. Having a bad first dot talk is not even unusual, as I am fairly rusty after summer and most of the time it's my students' first experience with them also. Surely, they will get the hang of it for day two!

Or.... I can pick easily the most confusing dot configuration on the planet, and then I died.

Do you see how arrogant I was when I chose this configuration? After the pain of the first day, I wanted to use that familiar "5 from a die" shape to get the party started. Then, I quickly tacked on three more dots in a "sideways triangle" type shape. (Please excuse all these technical math terms. ahem.) When I stepped back, I saw what they would do, but I underestimated how painful it would be.

Here you go, in case you don't see it yet. Daveelah's way nets 10 dots, and Alex's way nets 8 dots. All 30 students were nonplussed. It was totally fine with them that these two got two different answers. They both looked right, we understood their explanations, so why not?

No need to panic! I'm a professional. I quickly asked their table groups of 4 to determine how many dots there really are. About three quarters of my class was convinced that it was 8 dots, including Daveelah who offered this explanation:

I was pretty sure they would see that the two dots in the middle were counted once in the blue box and once in the green box. We had just done a venn diagram, so we were on solid ground. Fail.

But don't worry, I got out a bucket of chips and asked them if they wouldn't mind terribly BUILDING this thing we had just done, surely they would see that they couldn't build the first one out of counters unless they had the two middle chips stacked double high. Fail.

In fact, it was so hard to get them to even engage in the ponderings....that I was actually getting frustrated. Not a good way to start your math program. So I let it go (after a very small lecture on what I expect when I ask them a question..."I expect you to think about what is asked...you don't wait for somebody else to think, you dive right in!"), praised dear sweet Abi for her excellent model, and slinked out to recess/yard duty, defeated by dots. We never looked back. As far as I know, they STILL think it's fine that this had "two right answers".  

"Do you see how it matches, mathematicians? Who sees the five part in our dot card? Can you show us where the five part is in Abi's dots? What is the other part? Can we see it in both places? Abi, we understand what you did!"

I have no pictures of the third day dot talk. It was bad. I'm sure you believe me. Here is day four's dot talk though, and it was perfect! A square of four dots, with one dot added on to either end, and we finally got some traction.

Now just like that (*snap my fingers) we are cooking with fire when it comes to our dot talks. I can actually remember why I love them so much. <3

Launching Writer's Workshop

My writing coach is aces! I wish I could keep her in my room all the time; she is so observant, so positive, so tuned in to what kids are doing (and not) in their writing. I learn so much watching her mellow, instant conferences as she moves effortlessly through the room. And when we meet after school, and talk about what we saw, she always has great ideas for where to go next. Really, I need to stop going on about her, or somebody will come take her away. I feel the same way about my principal, haha. Like, nothing to see here! Move along! But really, they are both fantastic and I live in fear that they will be whisked away for greater things. We already share Lisa the Writing Coach with another (huge) school, so I'm on eggshells with this one.

Anyway, we launch Writer's Workshop on the very first day of school, and by the 15th day we are doing our first publication. It's our way of introducing the whole writing process, beginning to end, before we go into our first full unit (Narrative, for us).

I also have this little area above my writing center, which needs to be populated with the different things students write ("things" is, obviously, a very technical term) but meaning letters, lists, stories, postcards, notes, etc.

 

The 15 days of launching lessons are pretty flexible. They're structured enough if you're just beginning to implement W/W in your classroom, it will support you in the day-to-day decision making that keeps the class moving. But there's definitely room for adjusting and making it your own if you've been implementing for years. I am going to say, no matter how long I've been doing this, I will always appreciate the road maps. I just don't have the band width to kill it for 7 subjects every day. I'll take all the help I can get!

And well...that's how I came to be doing a lesson on "lists" during the first day of school to launch W/W with my second graders. The launching lesson was to be for teaching the process of Think-Pair-Share, but, honestly, our kids have been doing a knee-to-knee T/P/S protocol with A/B partners and structured language support since Kindergarten. We can all do it with our eyes wide shut, so instead, I decided to link the writing to a shared experience.

First thing in the morning, to launch Daily 5/CAFE strategy (check for understanding), I had read the students the book "Friends" by Helme Heine. It's such a sweet story!

Later, when they came back from lunch recess, I read them the story of Mean Jean, The Recess Queen by Alexis O'Neil

 

We had just had recess, then we read about recess, so it was time to talk about recess. They moved knee-to-knee to tell their partners what they liked to do at recess. They had plenty to talk about!

 (Writing Coach Lisa...down on the floor listening to kids! <3 her!)

After they had a chance to tell their ideas orally, I introduced the format of a list. They told me about where they had seen lists before (when they go grocery shopping, at Christmas time, etc.) We referred back to the "Friends" story (this was now the third time we'd used this story on that day, so they were very familiar with the characters and the plot). I opened the book to different pictures, and asked them to remember what the friends did on their adventure. As they remembered, I recorded them on our chart, and pointed out how I was making a list (the title, "Things the Friends Did Together", the numbered items, etc.). Then I gave them a piece of paper from a list notepad (long and skinny, with lines on it) and they went back to their desks to write a list of "Things To Do At Recess".

 What terrific little writers they were! They got right to work, and of the 30 students, 29 of them followed the list format I had modeled. Awesome! Lisa and I conferred with writers, mostly about stretching out words and writing what we hear...they were very hung up on "how do I spell..."  There's a place for that, but it gets in the way of getting our ideas down, so we will continue to develop our sound it/write it strategies. For our mid-workshop interruption, where you highlight some positive thing you see somebody doing, to help direct (or re-direct, as the case may be) any students who are having a hard time getting started or are confused about some aspect of the writing, we highlighted two students who had made a title and used numbers in their lists (most kids had done both without prompting, based on the modeled lesson) and two students who stretched their words and wrote what they heard.

The second day they wrote about a special object. We read Knuffle Bunny in the morning and Ira Sleeps Over in the afternoon.

 

I cannot be the only person who has used one or both of these texts to introduce writing about a special object, but I will go on record to say they work, alone and together, perfectly to inspire a full page of writing from Littles, which I for one, will be using as a baseline for their second grade writing.

Graphing Activities

 On the first day of school, I take a quick picture of each kiddo using my iPhone. I use these for lots of things, like the birthday displays, parent/family gifts, and portfolios and displays of student work. I never regret having those little mugs at my fingertips.

On the second day of school, we used them for our first "getting to know you" graph activity. The inspiration for this activity comes from this free download that comes with ideas for questions to pose, graphics for each one, and some headings. It's very sweet, and much cuter than the hand written stuff I usually do.

The "big ideas" I wanted to get at with doing the graphs with them were:
1) we can organize our data in different ways (this free download focuses on venn diagrams and bar graphs) and this includes tables, graphs, and tally marks.
2) when we organize our data in different ways, the data is still the same. In other words, the number of tally marks should match the number in a table, should match the bar in a bar graph.
3) we can ask and answer questions about our data.

I combined the graphing ideas in the download with a Kathy Richardson activity from this book, which I still think is one of the best resources for doing math with Littles:

 
Each student is given one unifix cube. They have to put their cube into one of the bags. This question was about whether you like to do things inside, or outside, or both. There was no "both" option in the download, as this comes up in the way it's organized as a venn diagram. In other words, you show you are both when you put your mark in the center of the overlapping circles. But I wanted to do this activity first, and so I made a "both" bag and they chose between the three options.

 Then, we make predictions...which bag do you think has the most cubes? Which do you think has the least cubes? What would no cubes mean? What can we already say for sure about our bags? (ex: "None of them are empty, so none of them have zero." etc.)

 As each cube is pulled out of each bag, students make tally marks to go with them.

Then students put each set of cubes into a tower, and we can start to ask and answer question about our data. Which had the most? The least? How much did inside and outside have together? When we put them together, do they have more or less than the "both" category? How many more is it than the "both" category? This is the most successful way I've found of dealing with the "how many more" issue with Littles. It's sometimes done in using "clue words"....like, "how many more" means subtract.

But it's so much more complex than that for little kids. First of all, if you have a lot of English Language Learners (our school is two-thirds designated as ELL), the subtlety of the language is pretty brutal.  I mean, "How many altogether" means add, and "more" means add, but "how many more" means subtract? And this is further compounded by the fact that, left to their own devices to make sense of a situation, most Littles will actually NOT subtract to solve this problem. 95 out of every 100 kids I've worked with, with no instruction on what to do with the cubes, will actually not "subtract", but rather will "count up".

In other words, if I ask how many more students like to play BOTH inside and outside, instead of JUST outside, 95% of them do not think "14 - 7 = ____"....instead, they think "7 + ____ = 14". So, to capitalize on this (which, actually, this thinking is very algebraic, so I want to encourage it, and here is this context where it comes up naturally for them, so #winning) and to help them make sense of "how many more" situations, I don't talk about it being a "subtraction" problem, but rather a "comparison" problem.

Are these the same? (no)
How do you know they are not the same? (this one has more)
So, they are different? (yes)
We can count the difference. Who has an idea, how we might count the difference? (take suggestions)
Summarize their work:  If it's more, we can actually count how many more, by counting the extras. If they are less, we can count how man fewer by counting what's missing. This is very easy for them to access and accomplish when they have the two towers to physically compare the amounts.

 They got a kick out of adding their pictures to the venn diagram. Before they added their faces I had them hash out where they thought the different options lived. We labeled an "inside" circle and an "outside" circle, and they figured out where the "both" category would go. (Some speculated it would go outside the circles, but they were convinced by others that the overlapping part would be both.)

32 Love Stories, Waiting to be Written

I moved classrooms (from building, to an outside portable) and grade levels (from 3rd to 2nd grade). I love all ages of children...I also can't stand all ages of children haha! not even kidding. They drive me to distraction and wreck my heart all at the same time. So moving grade levels is a non-issue for me, and I do it often and willingly.

But.

Moving classrooms? Nothing charming about that. Nothing at all. I feel like I used to have lots of storage and I was moving into a portable with very limited storage. I had culled out a lot of belongings LAST summer, but did it again this summer. I try to keep my work stuff at work, so if I couldn't find room for it in the portable, it had to go. I have about ONE INCH of available space at the moment. I mean, if I want to bring in a paper clip, something has to go. But it's (fairly) organized and (almost) clean, so #winning and I'm moving on.

Here is what it looked like when I got it:

Here's how it looked when I left tonight, with 32 second graders showing up in the morning....

 I am standing in the doorway, taking this picture to the opposite corner. I have 16 double desks, in pods of 4 students, for a total of 32 desks. Right in front of us is the writing center table. It has buckets instead of chairs. The buckets are full of math pieces. The big blue bucket in the center under the table houses recess equipment.

Immediately to the right of the door is my desk. It's not much of a desk to speak of. There are two heinous file cabinets right next to the desk. One is full of my teaching picture books and mentor texts, the other is full of my professional books. Did I mention the lack of storage? Under the desk is a rolling file cart that has my emergency sub plans (one week's worth) and my regular sub folder.

Past the file cabinets are built in shelves and counter under one of two windows in the room. I am using the space for my class library. Straight ahead is where the listening center will be. The black shelves are reference and picture encyclopedias. The word wall has the first grade words already on. The second grade words will go up gradually in a lavendar color.

 The book boxes for Daily 5. I have no idea if I'm doing these right. But it said to have them ready for the first day of school, just with a variety of several books. Also included, a folder, a word work notebook, and a reading composition book. Whisper phones and pointers on top shelf.

Continuing along the back wall, the "CAFE" bulletin board. On the right side are 4 clips for holding reading anchor charts. PS I love my alphabet. I found them tucked away when I moved classrooms, and I think they are just about perfect.

 My happy place. The bulletin board above the math pieces has clips for anchor charts. Puffy rainbow pom poms? Swoon.

 Still to go...a small carpet area with pillows for reading and problem solving, and the green bulletin board is intended for samples of their math recordings.

Next to the reading area are two tall shelves that came with the room. They were added in by the teacher who was in this room. What we see here are a lot of manuals and teaching resources for the adopted materials of my district. Behind the shower curtain (you heard me) are binders and some extra folders/notebooks. If I ever get around to cleaning that shelf up, even as minimally as the one next to it, I will take down the curtain. I do not hold my breath. Nor should you.

 The other window in the room, directly across from the first window. Sink is in the corner, and it has hot water! I've never heard of anything so luxurious! I made the curtains (from sheets) and the kids made the melted crayon art last year. The three pieces say Learn, Create, Laugh.

Down the counter from the sink, these clear boxes will hold math and literacy activities for independent work, the black buckets will have word work activities for Daily 5, there's a pink basket for some independent science explorations, and way at the end there, right before the one and only closet, is a box for student files and the pencil sharpener. Under the counter are some open shelves (with file boxes of math (of course! haha! never enough!) and some science and social studies units) that I covered with curtains made from same sheets as the window curtains. I used small tension rods to keep them in place.

 The reading table. Instead of chairs, they will sit on these buckets. ($5 with lids at Big Box Home Improvement Store and full of....more math!) Behind the reading table is my Science bulletin board, a shelf that has my word work centers and activities, stored until it's time to put them into rotation in the black buckets. On top of the shelf are some jars of incentives. The wicker drawer set was a hand-me-down from Pottery Barn and has all my office/student supplies in it. On the left are the student cubbies and a portable cupboard I brought with me from my old room. It has a lot of paper products in it.

 Continuing back along the front of the classroom...my whole group carpet area. This room had no carpet, so my principal got me this brand new one. Thank you! <3

Almost back to the door around this side of the classroom (as seen by the blinding light) is my writing center (in the cubbies) with a space for anchor charts over the center and the very simple calendar space partially covering the whiteboard. Above the writing anchor charts is an "I Can Write..." display prepared to showcase all the ways kids can write. The first purple square is going to be "Lists", which we will introduce and begin writing on our first day of class.

To the left of the writing, and just before the door, is the homework station. Folders are in the wire basket, and on the under shelf are leveled books for practicing at home. Directly above this is a very big, very empty, Writer's Workshop bulletin board that will hold their published works.

And that's it! I don't hate the portable. I am not in love with the dearth of natural light (I think I turned my lights on in my old classroom twice in 3 years, what with the wall of windows and the transom windows at the top of the 20 foot ceilings on the other side). I am quite smitten, however, with the walls. I HATE putting up bulletin boards, and I really really REALLY hate putting them onto the odd shaped boards of my old class. In the portable, the walls are all bulletin board material, so I could just make them the shape of the paper.

It's Dew Moss

It was a long week in writing training, but I'm glad I went and, after several weeks of planning and giving math trainings, I was only too happy to kick back and gather from others for a change. Our training was put on by district colleagues, and they did a fantastic job. In between writing lesson plans in my head, I learned a few things.

One of the things they had us do was to spend an hour each day doing Adult Writing. It was run as a Writer's Workshop, so folks who weren't around when the Teacher's College came out from New York and spent a summer training us on the Lucy Caulkins Writer's Workshop (um...jeez....6 or 7  years ago?!? Really???) could see the general format, content, and structure of a typical lesson.

In our math trainings, we also ask participants to do adult level math. It's important, as teachers, to experience the content as learners, too. It changes the way we approach it with our students, as educators, when we've struggled with something first at our own level.

In that spirit, of being a learner and a teacher of writing, I pushed myself to participate in the share at the end of the week. Today, five of us stood up and read our opinion pieces in front of roughly 150 adults. The piece I shared was imperfect, and untitled, but I was trying a strategy I had learned during adult writing (specifically:  use mini stories as evidence for an opinion claim).

Here it is, as I shared it today. It's about my son Tommy, who turned 13 this week. And, I'm sorry to say, it is an entirely true story:

(This is an untitled opinion piece)

As both a teacher and a mother, I can say with conviction that helping a child develop their own sense of a moral compass is one of the most important jobs we have. Children, of all ages, must understand the importance of telling the truth, even when doing so means that there will be consequences.

When my son Tommy was in kindergarten, he fell off the jungle gym at school. The two bones in his arm snapped like twigs just above his wrist, and he wore a hard cast for five weeks on his left arm. That was a lesson in gravity he never forgot. It was also the year he learned the importance of telling the truth, even when doing so means that there will be consequences.

It all started on November 9^th, 2005, when he reluctantly shared a note from his teacher. “Tommy was very disruptive at lunch time today. He first told us he didn’t have one, and then we found it hidden in the closet. His response: “Darn it!” Then he was swinging his food around and holding his sandwich in his mouth while he flipped his head. Please talk to him. ~ Mrs. McCallum”

Well, I’m sure for many people this would be dismissed as less than a tragedy, but I wasn’t going to let this go without letting him know how important it is to always tell the truth. “Tommy,” I told him, “It is very important that you do not lie. It’s okay if you want hot lunch ~ just tell the teacher that you have a lunch, but you want hot lunch today.  You don’t have to eat the lunch mommy packed you…but it’s not okay to lie. It’s the one thing we will not tolerate!”

I was confident that I had done my duty as both a parent and a fellow teacher (I got your back, Mrs. McCallum!). That night on the way home from soccer practice, we had a conversation in the car that started with Tommy telling me that some kids were calling him a jerk and a dumb ass at school that day.

What?? I said

Yeah

Are those nice words, or are they bad words?

He said they were bad words. Bery, bery, bery bad words.

My teacher antennae went up, and I had to ask him…”hmmm….were you calling them names before they called you names?”

No, mommy.

Well…you don’t use those words, right?

Right, mommy.

On November 10^th, 2005, we got another note from Mrs. McCallum:

“At recess yesterday, Tommy was calling other children ‘dumb ass’. When we asked him why he was using such words, he said, ‘My mom and dad call me dumb ass all the time.’  Please talk to him. ~ Mrs. McCallum.”

Now, look. We had a wild-acting dog that we definitely called a dumb ass. And that guy who cut me off in traffic? Well, I’m going to call him a jackass. But I categorically deny ever calling any kid a dumb ass.  That right there? That was a lie, and we must always tell the truth, even when doing so means that there will be consequences.

“Tommy!!! Mommy and Daddy do not call you that! You lied to me yesterday! You said other kids were saying it, but it was you...then you told your teacher that we call you that at home, and you know that's not true. Just because you were trying not to get in trouble! Well, Tommy, you're in a lot more trouble for lying about it. It's better if you just tell the truth, son, even if it's uncomfortable or scary, because lying is so much worse.”

The rest of that week was very hard on little Tommy, but I was sure that when the weekend came, he had learned the importance of always telling the truth.  That Saturday, we went up to Sacramento to a small park with carnival rides for a family friend’s birthday party. Tommy was very excited to visit with Porter and go on all the rides. Unlimited rides! For two hours! Rides, mommy! There are going to be rides!

We went to the booth to get his hand stamped for the rides, and that’s when we were informed that, due to Tommy’s arm cast, he would not be allowed to ride any of the rides that day. I’m sorry, but it’s very clearly stated in the rules!

Oh dear. Without saying a word, he turned and leaned into my legs, and he cried. Not a temper tantrum, not a wailing cry. It wasn’t big crocodile tears, or the manufactured drama of a child who isn’t getting what he wants. It was just a very subtle, very heartbroken cry.

So I did what any mother would do.

I told him to lie.

And not just lie! I put his jacket on, and explained to him how to keep his left arm completely inside the arm of the jacket ~ “don’t let your fingers poke out!” I told him ~ and we practiced how he should hold his tickets with just the tips of fingers, in the most natural manner so as not to call attention to his injured arm. I went on every ride with him and I helped him lie his butt off to every teenager working in that park.

It was an emotional week for Tommy, but he learned some very big lessons.  Most importantly, he learned that you must always tell the truth, even when doing so means that there will be consequences…..except…well….sometimes, he learned….you can lie.

 I showed Tommy the actual notes from this incident. His response when he read the part that said, "He told us my mom and dad always call me dumb ass"? He gave me the most perplexed look and said, "Why did you call me names?"  Oh, for the.....we didn't! YOU LIED! He legit couldn't compute it haha! He just gave a little shake of his head and furrowed his brow like we were the worst parents ever. Too good!

Writing

I've been at a training for writing in my district this week. I've been stealing time at the lunch hour and for a couple of hours each day after the training to work in my classroom. I'm pretty sure this is the time every year when I realize my to-do list isn't going to get done...moving classrooms, switching grade levels, and working the last month with trainings....I think it's time to put it all down and start preparing for 32 new love stories to be written.

Think Addition to Solve Subtraction

I wrote about combining procedural with conceptual knowledge using John Van de Walle's idea of ten-frame cards for addition facts just a few days ago. There's a link in that post for a set of flash cards with ten frames on them, and suggestions for how to use them.

This set (that you can also download the PDF for free by clicking here) is a little different. Van de Walle suggests using these to count up through ten. Seeing the ten frame with 7, for example, encourages students to think 3 more (to get to ten) then another 3 more (to get to 13). That makes 6. Easy!

I think one of the mistakes I make as a math teacher of young children is to move to "formal recording" too quickly. Letting them make sense of this counting up strategy, to solve a subtraction problem, is one of the biggest ideas they need to develop in elementary school. This falls under the umbrella of inverse operations...not just telling us that addition is the opposite of subtraction. Every 1st grader in California can tell you that. But actually using addition to solve a subtraction problem? Well that takes some finesse.

What does it mean when students are fluently, flexibly, and comfortably using these cards to count up to solve subtraction? Well, they aren't counting on their fingers. They are using the relationship to 10 to count up in parts. How many to get to ten? Now how many more to get from ten to the start number? And this can be recorded in the following way:

Another activity could be to have the students match a subtraction fact card to the corresponding addition card. For example, a student could match this 13-7 card to the 7+6 card in the other set.

Now, if I'm offering an organizational tip, I can just about guarantee it came from somebody else. This one is from Miss Peaslee (1999). She taught me that if I make 7 sets of cards, I should make each set on a different color of paper. Then, when students are working together with their cards, at the end of the day, you simply separate by colors and know that you have a complete set again. Yay!

Math Practice One: Get "UNstuck"

There are 8 Standards for Mathematical Practices in the CCSSM. They are the same for Kindergarteners as they are for 12th graders, as they are for adults and mathematicians. I love the very idea that, in theory, being mathy in kindergarten is essentially the same as being mathy in grad school.

One of the exercises we do is to make sense of the Math Practices in "Kid Friendly" language. It makes sense, right? The first few words of every practice is, "Mathematically proficient students..." so doesn't it make sense that they need to understand the practices? They are the ones who have to do the behavior. They need to own it. Unfortunately, the kid friendly language for this one can sometimes come across as cheerleading...You can do it! Never give up! Always try your hardest!

I mean, yes, these are necessary attributes of the practice. But they aren't the mathematical attributes of the practice. If you look at the details of the practice, we begin to see things like, "They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt." Let's be honest...there isn't a teacher alive on the planet who hasn't watched a kid try to add 4+112+2 to solve this problem: "A family of 4 drives 112 miles on 2 tanks of gas. About how many miles did they drive on each tank of gas?"

I just finished a week long summer institute, and it never fails that teachers inspire me to want to be a better teacher. So I started thinking about the things that I do to promote this practice in my students. Let's examine three "get unstuck" strategies from the practice.  Along the way, we'll put them in a more kid-friendly language, think about the classroom experience, and think about out what we can say to students to help push their thinking and behaviors around this practice.

From the practice: consider analogous problems 

Kid Friendly Language: "Can I think of another problem like this one?"

From the classroom: We want students to categorize problem types as they make sense of them. So once they know that "I have 8 stickers and my friend has 7 stickers, how many stickers do we have altogether?" is a problem where we are putting things together, they can begin to think about all the other problems they've done where we put things together, and they can think about and use some of the strategies that they used to solve those problems before.

What I say:  Start open ended...  "Can you think of another problem that we did that is like this one?" Add an optional more focusing question... "Is this problem like the puppy problem? Or is it like the pumpkin problem?"  Or how about a downright leading question... "Is this problem like the puppy problem where we joined them together? Or is it like the pumpkin problem, where we took them away?"

From the practice: They monitor and evaluate their progress and change course if necessary.

Kid Friendly Language: "Can I act it out? Can I try smaller numbers? Can I make a story? Can I draw a picture?"

From the classroom:  Part of persevering is having strategies for getting "unstuck". There is certainly an art to abandoning a current strategy if it isn't working, and knowing how to start on a new, possibly more fruitful, strategy. Kids don't often do this gracefully. They do, however, lick paper, roll on the floor, throw cubes at each other, take out a book, doodle, etc. So when we see students get off track this way in math class, it's a good bet that I'm looking at somebody who is stuck.

What I say: Always start with a question..."Can you show me how you've tried to solve the problem?" My next questions are usually based on what they reveal, but I'll usually refer them to the list of "Can I..." statements to see if there isn't a new way to think about the problem. If a student is just working through pencil/paper strategies and nowhere near a correct interpretation of the problem or accurate answer, it's not unusual for me to give them an answer. "So, if I told you that the answer is 15, could you show me with your blocks why that works?" or "So, if I told you that the answer is 15, could you draw a picture to explain how the problem works?"

From the practice: Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”

Kid Friendly Language:  "Do I get the same answer if I try it in another way? Can I convince a friend?"

From the classroom: You always have quick finishers, and students tend to over rely on a few strategies, whether they are efficient/accurate...or not. Suggesting that they try another strategy, or hooking them up with a partner who got the same answer to compare strategies, can be a useful differentiation technique. I've also hooked up kids with very different answers to try to convince each other that each is right. Good times, good times.

What I say:  "Andrea, Isaac just got a very different answer than you...can you two get together and see if you can understand what each other did, and come to an agreement on what's happening in this problem?" And when students ask me if they got it right I say, "Well, you've convinced me, but you also need to convince somebody else."

I made this little poster to use with my students. You can get a copy of it by download it for free by clicking Math Practice 1 - FREE (PDF).  Telling students "never give up!" and "keep trying even if it's hard!" does little if we don't also help them develop the tools that get them unstuck.

One final note: Never underestimate the power of the "take a break strategy". George Polya said, "A problem isn't a problem, if it can be solved in 24 hours." The end goal of any one problem solving experience is not necessarily "an answer". It's to develop strategies and mathematical understandings. That's not completed in a 50 minute math lesson. So go home, sleep on it, play soccer, do something else. Fresh eyes are the problem solver's strongest tool at times.

Clock Numbers

 I've been using these clock numbers in my classroom. I moved rooms this summer and have noticed that there is no clock in my new room.  Really? Once I get that hashed out, I have to reprint these and post them around my clock. I have to say, my third graders found these very helpful. I never even discussed them during the first few months of school, but even during that time, don't you know they started mentioning them to me? "Hey, Mrs. B...isn't it 10:15 right now?" or "Ohhhh, I see....that's like 4 fives, look, it's like counting by fives!" Stuff like that.

These aren't fancy, but they looked nice copied on astro bright papers, and you can get them for free by clicking here on Free Clock Numbers (PDF)  The file includes the numbers :00 to :60 and word cards for "quarter after", "half past", "quarter until", and "O'Clock"