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!

Spinning with The Pigeon

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

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

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

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

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

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

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

You can download from my google drive here:

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

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

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.)

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"

10-Frame Flash Cards

Click on the link to download free 10-frame flash cards....read on to find out what they are and why you need them ;)  Set of 10-Frame Flash Cards- Free Download (PDF)

 In John Van de Walle's book, Teaching Student-Centered Mathematics, K-3, he devotes a whole chapter (Chapter Four) to helping children master the basic facts. He writes that Mastery of a basic fact means that a child can give a quick response (in about 3 seconds) without resorting to noneffficient means, such as counting. He doesn't talk about memorizing here, but he does talk about helping students develop efficient strategies based on relationships between quantities (such as anchoring on 5 and 10  to master facts such as 8 + 6 by thinking of 8 as 10 with 2 missing and then getting that 2 from inside the 6 and making a new fact, 10 + 4) and relationships between operations (such as thinking addition to solve subtraction).

There always seems to be some controversy between the "memorize your facts" group and the "develop your number sense" group. But these things aren't mutually exclusive. Memorizing your facts is very helpful. But, developing your number sense is helpful AND important.  In the text series adopted my district, the lessons for subtraction in 2nd and 3rd grade direct students to "think addition" to solve subtraction problems. So for the problem 14 - 8 = ___ students are told, "Since you know that 8 + 6 = 14, then you already know that 14 - 8 = 6."

The only access for students is if they've already memorized 8 + 6 = 14. If they don't have recall of that fact, they have no other strategy for thinking addition to solve 14 - 6. I think this is why people confuse fact memorization as important, instead of just helpful. If it's the only strategy we develop, then it's of the utmost importance!

But think ahead now. Nobody can memorize every fact. I mean, if the child is solving the problem 54 - 28 = ____  is there anybody who thinks it would be reasonable to tell them, "Well, don't you already know that 28 + 26 = 54? " Of course they don't hold this information at a recall level.

But let's go back to the 14 - 8 = ____ problem. If we think addition that 8 + ___ = 14, but haven't memorized that 8 + 6 = 14, I can still derive this fact. I can think about needing 2 more to get to 10; then once I'm at 10, I need 4 more to get to 14. So now I've derived the missing addend as 6, but in a 2 part and a 4 part. Students (people) can do this so efficiently that it can appear memorized.

When students have ample opportunities to develop this strategy of counting up in parts, then they can also use it in the 54 - 28 = _____ problem. If we think addition as 28 + ____ = 54, we need 2 more to get to 30.... 10 and 10 gets us to 40, 50...and 4 more gets us to 54. So the parts we used to count up are 2, 10, 10, and 4, for the answer of 26.

The Common Core Standards are written so that students have both procedural and conceptual understanding. I guess I've heard people say that it's more important to teach procedures first, and worry about the conceptual understandings later. I've also heard people say that you have to develop the conceptual understanding before students can use procedures.  I think everyone must have their reasons for saying so, but in the reality of the classroom, I have found that it happens much more organically than that. You can actually teach the procedures and  concepts in any order or even at the same time. If you are providing a variety of opportunities for students, they tend to take what they need, and cleave to the important ideas, as it makes sense for them.

So here come the flashcards that John Van de Walle suggests.  An old-school flash card has the fact written, such as 9 + 7 and students have to recall 16. But adding the 10-frame quantities to the flashcards help students develop their number sense too. I can SEE now that 9 is 10 with one missing, and I can VISUALIZE moving one dot from the 10-frame with 7 to fill in the ten-frame with 9. Now I'm using the Make a Ten Strategy.

 

Or now, instead of only having access to 8 + 7 if I memorize the answer as 15, I can now use the 10-frame quantities to see that if I just cover up that last dot on the 10-frame with 8, I've created a 7 + 7, and I only need to add back the covered dot to get the answer 15. Now I'm using the Use a Double (Double + 1) Strategy.

Alright there are your procedures right alongside your conceptual understanding! You really don't have choose. Other ways I've used these cards is to have them sort the cards by which strategy they would use...for which ones would you make a 10? Which ones would you use doubles? Which ones make the most sense just to count up? Which ones do you know "by heart"? Can you explain to a classmate why the strategy you are using makes sense for these numbers? Are there any cards you would use more than one strategy for?

Ohhhhh the possibilities are endless :)

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!