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*

First Problem Solving Workshop

We did our first problem solving workshop (PSW) on the 7th day of school. The first five days were introducing, exploring, and practicing with our tool bags.  Once that was settled, we were able to dive right in....by which I mean oh wait, wait another day because my schedule got rearranged at the last second. Oh, last minute changes, you vex me.

The PSW is my very favorite way to teach math when it comes to operations. The entire thing is based on my understanding of Carpenter/Franke's Children's Mathematics (Cognitively Guided Instruction).  If you teach any workshop, such as reading or writing workshop, you can use the same structure to create space for students to make sense of, and solve, problems.

The workshop structure includes a mini lesson, a work period (which may include a mid-workshop interruption), and a close.  The whole thing takes about 35 minutes - 45 minutes.  Shared below, as our very first PSW of the year, is a mini lesson (introducing a problem together, including reading and acting out) for 15 minutes; a work session for almost 20 minutes including clean up; and finally a quick close on the carpet, less than 5 minutes.

I have been doing this for many years, and I have always started with a "put together" problem. For the first time, I decided, at the last minute (oh, last minute changes, I LOVE you ;) to make the "separate" problem the first one. I am too pleased that I did!

This is all based on Carpenter's work, which I highly recommend that you read, start to finish.  Until then, the main points are 1) children can resolve math story problems without you and your snappy ideas; 2) they do this in totally predictable ways that they construct themselves; and 3) not all math story problems are created equally....some "problem types" are harder for children to access until they have built up a barrel of number sense and an understanding of part-whole relationships.  In reference to #1, if you start with the most accessible problem types, you won't have to do any underlining of key words or circling of numbers or any of that other stuff. And that's because of #2, which includes the information that virtually all students (when we stop pecking at their necks) begin all understanding with a strategy called "direct modeling".

In direct modeling, students will act out, with manipulatives, exactly what the story says, in the order it says it.  So if the math story says "I have 3 cookies. My mom gives me 2 mores cookies. How many cookies do I have now?" a child will read "I have 3 cookies" and place three blocks down. Then the child will read "my mom gives me 2 more cookies" and place two more blocks down.  When they read the question "how many cookies do I have now?" they count all the blocks they put down. (Worried that you have students who can't read this problem? Me too! I won't bore you with the results of my Fontas and Pinnell assessments I just completed, but please know that I have exactly 7 - out of 29 students - that would be considered on "beginning of 2nd grade level"....it's a real thing.)

Which is why, whenever we start a new problem type, we always start with a group reading and acting out of the problem. I like to use names of my students, and their real hobbies, to get us started. In this case, Ishika likes to collect rocks and sea shells, and her class buddy is Jaiyana.

The process is simple: I read it to them from the poster (with blanks....I say "hmmm" for the blank...Ishika has hmmm shells, like that).  We read it all together. I point out that not everyone is reading, so we read it chorally again. Except I have to interrupt us after the first sentence, because not everyone is reading. So we try again. Rinse and repeat UNTIL they GET it that EVERYBODY will participate, full stop. Once we've read it, I have them partner talk about what's happening in this problem. How would they describe what's happening?

This time they shared:  Ishika has shells. (What is she doing with the shells) She's giving shells away. (to whom?) She's giving them to Jaiyana.  (Who is "she" in the last sentence? Is it Ishika or Jaiyana?) It took some discussion, but they decided it was Ishika. Those referent pronouns with second language learners (21 of 29 are, for my class) can be tricky. It's worth talking about.

I gave them the black number pairs first, and taught them to populate the blanks with the numbers, in order. We did 8, 3 first and re-read it chorally as "Ishika has 8 shells. She gives Jaiyana 3 shells. How many shells does she have left?"  Ishika and Jaiyana come up and act it out with cubes. We prove the answer is 5. Then two other people play Ishika and Jaiyana and we did it again for the numbers 7, 2. Two other actors did 9, 3. And a final pair of actors did 11, 2. They would have gone on, but really, it was enough.

As you can see in the picture above, teaching this system of reading the problem with number pairs is extremely helpful as you try to differentiate in your classroom. For the student who finishes in seconds, you can quickly write two more difficult numbers on a post-it and tell them, "These are your new numbers" and they get right to work on it. Likewise, when you come across a Little One who just can't get started, who can't get any traction, you can quickly jot down two smaller, more accessible numbers and say "Try it again with these". Since you taught them how to populate the problem with any two numbers, they are good to go.

When I send them off to the do the problem on their trays with their bag of math tools, I sent them off with the red numbers, 22, 7.  We read it altogether one last time, and then we go to work. Super important: I always send them off with numbers that are just out of their reach....I want it to be a problem they haven't memorized as a basic fact, so that they will rely on the direct modeling intuition to solve the problem. In this case, I also want them to struggle with dealing with the notion of needing to regroup....If they build 22 with two ten-sticks and two units, they are going to be confronting some big math ideas right away. Yummy, I say!

 My job during the work period is to move through the room gathering information, and to do what I call "match making"....finding pairs of students who should talk to each other. It's always the same at first, "Teacher the answer is 9" and that's immediately followed by another breathless student saying "Teacher the answer is 14" at which point somebody will say "No teacher isn't the answer 15?" and I will say, simply, "The three of you need to take your trays to the carpet and explain how you did the problem....do you agree with each other? Is there a way to prove which one of you has the right answer? Do any of you have the right answer?" And so on, until just about everybody is partnered up and discussing the mathematics.

The other job I have is to gather up which strategies they are using. Are there any that should be shared and promoted because they are efficient, flexible, or mathematically important? Are there any that make me want to cry? haha Not even kidding about that one.

 

This is a really common way of building the 22 in the beginning of second grade. This child is not yet thinking in tens and units, just counting all out by ones.

 This child is also counting all out by ones. The ten sticks are not being thought of as "groups of ten", he is counting each one as "one" and needs 22 of them to make 22. This is less alarming than you might think....lots of students interpret this tool as a "tally mark" almost, rather than as a group of ten. They will make sense of it shortly. In the tray next to him, some enterprising lovely has made 22 as two ten sticks and two units. Thank you, sweet one, this will be very helpful, very soon.

Also very common, is the really long train method of making any number. This is a version of thinking in units, rather than thinking in tens and extras.

That's it! Isn't it lovely? There's no real resolution at the end of these first session, but we have LOTS of information about how our kiddos are making sense of quantity and subtraction, and we have LOTS of great conversations and proof.

At the end of this first session, we clean up and meet back down on the carpet. So much math to share! But I hold it until the next day, when I can craft a sensible mini lesson out of all that math to start our next work session. For today, we close with a quick debrief on the way this works:  We solve our problem, and we get an answer....BUT....we are not "done" until we have talked about what we did with somebody else. Do we understand them? Do they understand us? Are we in agreement? Here is where we can make the point that talking about our math is what let's us test and prove our ideas.  Here is well I will also, gently, suggest that you can talk to each other without first asking me...find somebody to work with, you won't be sorry my Littles!

First Week of Math - Exploring Tools

Over the first 5 days of school, we introduce, explore, and divvy up a set of math tools that the students will use during problem solving workshop for the whole year.  It starts on the first day of school. I start number talks on the second day of school, and it's always a "dot talk". This means that I need them to play with dots to exhaustion on the first day of school so I can protect my sanity when they need to use the dots on day two.

The first step is to lay down some ground rules. I'm not going to lie...this is a loud, sometimes crazy and explosive process.  My philosophy here has always been, "It's exciting to get an idea and test it!" As far as the students are concerned, they are pioneers!  Nobody has EVER thought to make a train of cubes that goes around the ENTIRE carpet before (everybody has thought of this haha but they don't know it) and thinking of this then doing this can be very exciting...which requires some hopping...and shouting.  I do try to minimize this, but I also ignore a lot of it during this free exploration period.

Before we start, we brainstorm what it "looks like" and what it "sounds like" when we are being responsible with the math tools. (They don't go in our mouths, up our noses, no throwing, we share, we ask if you are done with those before we grab them, etc.)  After we finish, we reconvene on the carpet and go over how we did....did it "look and sound" like the important and serious work that it is?  Here is where I will remind them that we should be using "level 2" voices, which is a normal voice, but not an outdoor voice.

Finally, before we walk through this, may I suggest a "shout out" call to attention?  Bells, chimes, sing songs....I do it all, as one must if you are serious about having their attention when it matters, but I have to admit, nothing beats a call and response shouting when the noise level creeps up.  For the last five or six years I've been (literally) yelling "yo yo yo!" and they (literally) yell back "yo what's up!" then everybody freezes and we have a window of opportunity to make an announcement, call for civility, and/or transition out of exploration.

We follow the same process every day. Set out a tray of one type of tool for each table group. Go! I walk around saying "yes" as often as possible. "Teacher, can I make the world's longest train?" Yes you may. "Teacher, can we work on the carpet?" Yes you may. "Teacher, can we have more red ones?" Yes you may. "Teacher, can I make a tower?" Yes you may. "Teacher, it got too tall, can I stand on a chair?" Yes you may. (Let me just stand here next to you though, how about that?) I also carry a clip board with this little checklist of observable behaviors, made by my dear friend and teacher extraordinaire Kristy. It helps me focus on the math that is happening amid the chaos, and it also lends an air of officiousness that is sorely lacking without it.

At the end of each work session, as a new tool is cleaned up and put to bed, each student gets their math bag (students are assigned a number that goes with their name - "a" kids start with number one and end with "z" kids at *cough* 32 when I'm lucky) from the hook and counts out a number of the tool to add to their bag.

(The tool bags started as gallon ziplocks, but I eventually repurposed a few cheap sheets and maxed out my sewing skills by make three straight lines for the seaming and one hem for the drawstring. The dimensions are still roughly those of a ziplock gallon storage bag, though taller....13" tall by 11" wide, give or take, finished dimensions.) (ETA:  Each bag has a number written on it in fabric paint, which matches the numbers assigned to the students, which matches a number on the wall behind the bag, not yet affixed in this picture.)

 At the end of this period, each student puts 25 red/yellow counters in their bag.

 At the end of this session, they add 40 unifix cubes to each bag.

 

 At the end of this session, they add 40 ten sticks (not 4, worth 40, but 40 actual sticks) and 25 units.

 At the end of this session each child adds 60 flat square tiles to their bag.

Other things that will no doubt likely end up in their bags include an expo marker, a die, and some hundreds flats.  In the picture below, you can see that MORE of each of these tools is available in this open storage. As we begin to solve problems using our tools, we may find ourselves in need of more of one thing or another, and we can always come here and get them. That basket of hundreds flats is always available too, but doesn't usually gain popularity for some time.

  On the fifth day, we don't introduce a new tool, but we practice taking our bags out, using the trays (seen above stacked up on the side) and practicing using our tools in whatever way we choose, but focusing on keeping OUR tools on OUR tray. After cleaning up and putting everything away, we also introduce the "I Found This!" bucket, seen on top of the stack of trays below. When we find ANY math pieces, at ANY time, on the floor, kicked under furniture, inside our pencil boxes, any old where at all.....we put it in the "I Found This!" bucket (named thusly by the number of students who walk up to me and say "Teacher I found this").  Make sure you are clear....no puzzle pieces, crayons, broken pencils, hair (!) goes into this bucket. Only math pieces! At the end of each week, it's somebody's job to sort everything back into those open buckets.

I used to spend time, every summer, counting out the tools for each child. Sometimes, I even had them in little plastic snack bags INSIDE the math tool bags. But then, to be honest, I was wayyyyy too invested. I needed to just divest from the entire process. When I tell them to count out 40 or 60 of something, in all likelihood, some number of kids will miscount. Don't care. Some number of kids will eventually find that their bags are empty and/or a certain tool is totally not accounted for. Don't care. That's what the open storage is for, go get some more of whatever you need.  But as you can see, by the end of the year clean up photo below, somehow it all ends up back with me. It's a zero-sum game from my perspective....they go out, they all end up back. What happens in between, I had to tap out. I have enough crazy to manage without adding "what's in your tool bag" to the list.

I'm including this as your end of the year clean up tip, even though I'm assuming you are MUCH smarter than I am. This system works...Each table group gets a small bucket for each tool type. They sort their own bags into those buckets. Those buckets get dumped into the five (10?) gallon paint buckets we use for chairs at our reading table and writing centers. Kids who get done early can start making ten trains out of the cubes, since they store better that way.  Every couple of years we take buckets of tools out to the playground and add a bit of dish soap and water. When they ask why we are cleaning the tools, I tell them it's because they like shiny things. They agree, and we carry on. Love second grade :)

Here are the kids the first year I taught second grade, when we dumped ALL our bags on the carpet at the end of the year and tried to sort them into the buckets from there. It took four days, and they had lost interest long before any serious headway had been made.

 And here's me, on day two of four.  Just saying.

A Little Help Goes A Long Ways

One of my favorite number routines is, of no surprise, the Number Talk. But perhaps my SECOND favorite is a "number of the day" exercise we do in our journals during our calendar work. We count the number of days we've been in school, adding a sticker to a ten frame each and every day.

Oh! Quick bird walk. This is not my idea, by any means, but it changed my life, so just in case you don't do it yet, here it is....quick and powerful, and it requires no brain power from me. Could it be more perfect? I used to have the calendar set that had the straws? Where you'd count the day, and add a straw to the little pouch labeled "ones" each day...then when you had 10 straws you'd bundle those up with a rubber band and move them to the pouch labeled with "tens". Nevermind, here:

Okay but many years ago I switched to this:

And never looked back. Never once did my kiddos ever say "We're almost at ten" when I used the straws. We just kept counting them every day. Over and over. But with the dots on the ten frames, they suddenly started saying it spontaneously..."Two more days until ten! We almost have another ten! Here comes another ten day...let's see...four more days, and it's Thursday today....so we should have ten on Wednesday." I do not jest or exaggerate. And as for my teaching? I did exactly zero differently. These babies teach themselves. And, bonus time, when you fill them in side to side, you get to talk about odds and evens every day because the dots have a partner...or they don't.


Okay, back to the point. Number of the Day is another math routine that I love a lot. It's perfect for getting kids to use the properties of arithmetic...like commutative, associative, identity...you know, Big Ideas. Each day, the number of the day is the number of days we've been in school. We start with, "You have to make that number five different ways"...On the tenth day of school, you have to start making the numbers ten different ways....on the twentieth day of school, it's 20 days. Kids find patterns (plus one/minus one compensations for addition and minus one/minus one or plus one/plus one compensation for subtraction) plus what happens when you add/subtract zero, and whether the commutative property works for addition AND subtraction...it all comes up, even when you are just trying to make the number 1! Such a boon, I love it.

Here's sweet, hard working Alex. I spot check the journals the first few weeks of school (and then every few weeks after that, as comfort sometimes breeds laziness in these activities), and this is what I found in Alex's journal as he was recording the ways for day number 14...which means, he should have been making 14 over and over.

Oopsy! That won't do! So I showed him how to use unifix cubes (and keep getting another one for each new day, to add to his train) and then we practiced breaking the train into parts, counting those parts, and recording what we did.

 

He's currently recording it strictly as addition, but we'll work on relating what he's doing to subtraction as the weeks progress. Two great things came of doing it this way:

Here's his journal just a few days later, for day 17. {LOVE} And the second thing is a different student was having a similar struggle and I said, "Hey, Alex, why don't you show your strategy to Vineet?" So he did. And that moment, when he was showing somebody else what he could do, that  my friends, was a magical moment for The Boy Who Struggled.

 

Cheap Ikea Trays

I have had these Lakeshore plastic crafting trays in my room for the past 4 years, and they are going strong. They are $15.00 for 4 of them, which makes them prohibitive, in my opinion. At the time, I had gotten a wee bit of money for my classroom, and had purchased 2 sets on a hog-wild-whim. The remaining six sets (for $90 plus tax) were out of pocket for me. I loved them that much.

 Here they are in my math space, stacked up there on the left hand side. We use them very nearly daily, and we love them. Just you can see them in action here, and also, just about anywhere I talk about doing math with kids.

 

Whenever I'm sharing with teachers about doing math with little kids, these trays come up. They are so appealing! I used to use felt mats I cut from a roll of felt that was gifted to me. Those were free, and I used them for two years. (I cut them up and made them into whiteboard erasers since I retired them from being math mats, so I am very grateful for that gift!) The mats were good, because it defines your space. Nobody was allowed to touch your pieces once they were on your mat, so it created a nice culture in the classroom and kept others from "tidying up" your opus. It also helped with the management of 33 Littles, as they were able to move to any spot in the classroom (on the carpet, other work tables, a corner of floor) and they felt like (and acted like) they were at their own desks because of those mats. Here they are in action, with the plastic zip lock gallon bags that were our original "Math Bags".

 Then I got the trays and I saw that, really, there were some serious advantages... Case in point, this is J, and she wanted to show me this great idea she had. So she trotted it over to where I was, which she could not have done on the flimsy felt.

 Because teachers are so clever and resourceful, I've heard of lots of great ideas for having the "tray experience" without the outrageous price tag. The dollar store sells cookie sheets, for example, which I thought was super smart. There is only reason I was STILL glad to have my fancy trays after hearing that awesome idea, in fact, and that was because my trays are extra deep compared to the relatively flat lip of a baking sheet. That's not a deal breaker (especially for the cost savings) but there are many times when we have to set our math aside for the day, and the extra deep sides of the Lakeshore trays means that we can leave our math tools set up and still be able to stack them without worrying that they would tip over or not stay flat. Super helpful!

 But just this weekend I was at Ikea and found these Smula Trays. They are a frosted plastic, nice and deep, and almost exactly the same size as my trays. They are a nice hard plastic and I love them. And, they only cost $1.99 each! What a bargain for what I think is a very perfect math tray!

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!

More on Math Tools

I get so many questions on how to use math manipulatives when I do professional development on teaching math. I wrote about many of the logistics in this last post, but really, I'm sure I will continue to write about in various forms and contexts. It's that interesting, I promise! ;)

I started out explaining that I present the groups with a tray of a certain tool, and they are given free exploration time. When I was a math coach/specialist for my district, I actually had this process written into the pacing guides. So the first five days of every school year are about introducing tools and procedures, including number talks.

Because I have an abundance of tools (largely scavenged from hallways and supply rooms where they were discarded) I just give everybody the same tools. If I were short of supplies, I would fill each tray with something different and rotate them over the five days. That's still legit for many reasons, actually, but I choose to do it this way because they also learn about the tools by watching each other use them, and talking to each other about what they see happening in different groups.

While they are working with the tools, I am circulating and learning a lot. I make notes on conversations I hear, math I see, and procedures I may (will) need to reteach. You can learn a lot about who your kids are in math class by watching them work in free exploration.

Who are your sorters...

Your artists...

Your pattern seekers...

Your hard core playahs...

Your spatial geniuses...

Your future engineers....

and, of course, Your Mopers

 I'm always surprised at how much math surfaces during these play exploration sessions. And if it looks like play? So be it.

Since they have their tool bags (these are last seasons bags, and the new ones are much roomier for the same number of tools), they are in charge of deciding which tools to use, to solve which problems, and in which way. I'm not going to lie to you...the first year I implemented this, and I saw what they did, I thought I'd made a HUGE mistake.

For example? Do you see the 18?

No? Me either. Until he explained: 1, 2, 3, 4 cubes...2 green tiles...1 yellow cube...2 sticks...1 red counter. (4 and 2 and 1 and 2 and 1 is 10). Now look at the dice....one says 3 and one says 5. 10 + 3 + 5 = 18.  This is certainly not what I envisioned!

But we pushed through this part, and it has become easily one of the most powerful, sense-making policies of our math period. It took some time, but we always highlight strategic use of tools. I'll often ask students doing something efficient and/or interesting to tell us a little bit more about why they did it that way. "Why did you use a 10 stick instead of the red counters?" Or, for a problem involving regrouping for subtraction: "I see you used a stack of 10 unifix cubes...how did that help you do the subtraction?"

It also takes offering an invitation to experiment. "What if you used _____? Would you get the same answer?" The more students used the tools as part of their daily practice, and the more invitations I offered to do the same problems using multiple tools, the more the kids made sense of the tools and they began to actually use them strategically. (Standards for Math Practice #5, CCSS)

Using, Storing, and Managing Math Manipulatives

 I am passionate about teaching math, most especially with Littles. I myself love math. A seven year old is a very imprecise tool with which to do math, and can be particularly baffling, for a lover of math. But the thing is, is that the most profound mathematics are introduced and mastered with 5, 6, 7, and 8 year old kids. The number system, relative magnitude, every operation, geometry, relationship and comparisons,  problem solving and logic...it's the same math that underpins algebra, trig, and calculus...it's the same math that underpins engineering, architecture, and balancing your checkbook. It's deep, important, and complex.

Students have access to these tools at any time. They can come here and get a "scoop" of counters or a stick of unifix cubes, anything they think will be helpful...even when I don't believe it will. True story, I bite my tongue a lot.

The new Common Core Standards are, necessarily, imperfect. But if they do nothing else, they elevate the DOING of math to the importance typically reserved in elementary schools for the LEARNING of math. Math so often happens to kids. They are expected to learn math by watching teachers do math. They are often taught discrete rules and procedures in order to get them through arithmetic, without ever developing their sense of being mathematical. When I was a math coach/specialist for my district, it never ceased to amaze me how little cognitive lifting kids were being asked to do in math class. Teachers, pressured to the point of breaking by NCLB (No Child Left Behind, or as we call it, No Teacher Left Standing) requirements, often shied away from problem solving and cleared a path through story problems wide enough to march their whole class across without ever giving kids room and space to struggle and grow mathematically.

I ask myself: would I try to teach reading by never giving a kid a book to read? To me it is the same thing. I can't fathom trying to teach math without giving kids a problem to solve.
 

I moved into a classroom with very little storage this summer, and I invested in these white buckets ($7 with lids at Home Depot) to provide additional math tool storage and seating at our writing center (shown) and reading table.

The eight Standards for Math Practices all start with the words mathematically proficient students to describe how students (people) behave when doing math. The fifth Math Practice is about choosing tools. It's not just about breaking out the base ten blocks to show kids how to make exchanges for the regrouping procedure in subtraction. In fact, research has shown that when teachers decide which tool to use, and how to use it, the students experience the tool as one more set of rules they have to memorize...instead of contributing to the true goal of conceptual understanding.

Making sure my students have access to a variety of tools, as well as promoting a culture of understanding where students are asked (no, required) to pick their own tools and make sense of those tools in a variety of contexts...this is no small undertaking. The pictures above show how I store the math tools.

In the beginning of the year, we spend time introducing the tools. Partly to make sense of them, and partly to outline expectations. We do a "what does it look like and sound like when we are using our math tools" anchor chart.

During the first five days, I put out a tray of different manipulatives each day on each table group. Go to town! We focus on behaviors and procedures for sharing and putting away the tools.

Finally, when they have "met" the tools, the are each assigned a "math bag". Over the years that I have been doing this, it has evolved from a gallon ziplock back stuffed into an already crammed desk to this arrangement, which I'm calling AWESOME. I used a sheet to make 32 drawstring bags. Nothing fancy. Cut rectangles out of the sheets, folded over the top to make a casing for the drawstring ribbon, then folded over the rectangle in half and sewed along the bottom and open side (don't close the top, you have to feed through the ribbon). PRO TIP: once you've run the ribbon through the casing, bring both ends together and tie them into a single knot. You are welcome! ;) These were considerably smaller in past years, I just this summer redid them. Improvements: they are bigger than the last ones, by twice.

 Also improved: I used little 3M tape mounted wall hooks to create a storage area under the whiteboard. It was fun when the bags spilled out of their desks and pieces flew across the floor two to six times a day but, alas, all good things must end. Each bag space is labeled with a student number (1 to 32, I number my students from the time they walk in the door. With a 35% mobility rate, we remain in alphabetical order usually no later than October. By then there's been movement, but when you come in, you just get the last person who moved's number. It means Montse Rivera is going to end up being number 2, ahead of Luis Chavez, but pay that no mind. Number order please!)

 The labels on the bag space now match the numbers I put on each bag. I used puffy fabric paint that comes in a tube and I freehanded the numbers. Because I'm a gangster like that. haha

Students use these bags during the math time. If they need more or different tools than what are in the bags, they can get it from the bins or buckets. Like Crystal here, who was counting out 60 pencils for her story problem...by counting each 10-stick as one pencil...at the beginning of third grade. Ayyyyy....my aching head.

 When Crystal told me she needed more ten sticks, what I thought was, "No you don't. You need 6 of them." What I said was, "Sure, they are over there." And then I made a mental note that half my class was just like her, and I had some place value work to do. Ahem.

In my second grade class, each math bag has: 40 ten sticks, 25 unit cubes, 25 red and yellow counters, 40 unifix cubes, 40 square tiles, and a die (for centers and games). I'm pretty sure it's exactly the same thing they had in third grade last year...maybe they had 60 square tiles.

Also available in the bins and buckets: more of all those, plus hundreds flats, thousands cubes, cuisenaire rods, fraction pieces (strips and circles), atrribute blocks, centimeter and inch cubes, plastic coins, and pattern blocks.

Available at all times in baskets around the room: rulers, measuring tapes, protractors, ten frames, and balance scales.

It's a process. We go back to review/reteach how to store and take care of our manipulatives regularly. But of all the things I've done in the last years as a teacher to open up the world of mathematics to my students, this is up at the top. So worth the aggravations and inconveniences!