# Can You Build It?

This week I’m starting to do a little math with my daughter everyday to dust off the cobwebs before 4th grade starts in September. One of the resources I’m using is the centers from the Illustrative Mathematics K-5 curriculum (Link to Kendall Hunt’s version of IM K-5 Math).

We kicked things off on Monday with a center called Can You Build It? (Link) One thing I like about the IM centers is that they often contain multiple stages within the same center, so you can choose just the right starting point within a given concept. Since my goal was to revisit arrays and the meaning of multiplication, we started with Stage 1. In the original IM version, one person builds an array secretly and then describes it to their partner and the partner tries to recreate it.

I changed this stage into a cooperative game that turned out to be really fun for my daughter. Here’s how it works:

1. Draw a target area card. (I created a deck of cards that have the numbers 10 – 27 on them. This means there are 18 possible target areas, which feels like a good range. The numbers are also small enough that you won’t spend all your time counting out the tiles you need before making your array.)
2. Each player secretly makes an array with that target area.
3. Share your arrays. If you made the same array, you collectively earn 1 point. If you each made a different array, you collectively earn 2 points. (To clarify, a 2 by 6 array is the same as a 6 by 2 array.)
4. Earn 5 points in as few rounds as possible.

If you don’t have square tiles handy, you could use a free app like Number Frames from the Math Learning Center (Link) which can be used in a browser or downloaded onto a tablet.

Or if you still want something hands-on, you could always use some crackers!

After a couple of days playing Stage 1 and revisiting how to build and describe arrays, we moved on to Stage 2. There are a couple of key differences here:

1. Instead of secretly making only one array, the goal now is to make as many different arrays as possible with the target area.
2. The game is competitive now. The player who makes more arrays earns 2 points and the other player earns 0. If both players make the same number of arrays, they both earn 1 point. The winner is the first to 5 points. (The original IM center used a slightly different scoring scheme. I opted for something similar to the game we played for Stage 1.)

My daughter immediately started bumping into ideas related to prime numbers. Here are some highlights from our conversation as we played for the first time:

1/ Daddy: Today our game is slightly different. This time when we draw a target area, our goal is to make as many different arrays as possible. If we get the same number of arrays, we each earn 1 point. If one of us makes more than the other, that person earns 2 points.

2/ Daddy: (draws card) Our first target area is 20.
(both make arrays in secret)
Daddy: Oh! I forgot that one!
Me: You have to remember you can *always* make a 1 by array!

3/ Daddy: (draws card) Okay, this time our target area is 13.
(both make arrays in secret)
Me: Ugh! I can only make one.
Daddy: Me, too. What did you make?
Me: 1 by 13.
Daddy: Hmm, I wonder why we could only make one array.
Me: Maybe because it’s an odd number.

4/ Daddy: (draws card) Now our target area is 11.
(both make arrays in secret)
Me: No! You can only make one again.
Daddy: Huh, is this an odd number, too?
Me: Yeah.

5/ Daddy: (draws card) Ok, our target area is 10.
Me: I’m just going to write down the 1 by array. I don’t even need to make it.
(both make arrays in secret)
Me: A 1 by 10 and a 2 by 5.
Daddy: Same here. Is 10 odd?
Me: No, it’s even.

6/ Daddy: You made two really interesting observations today. Do you remember what they were?
Me: …if a number is odd you can probably only make one array?
Me: …and you can always make a 1 by array for every number!

Originally tweeted by Splash (@SplashSpeaks) on August 18, 2021.

I love how this game has a simple premise – make arrays – but it creates opportunities for students to notice deeper ideas about numbers and multiplication. If you woudl like to try this game out with your own child or students, here’s a link to the center. (Link)

If you work in a grade level that introduces prime and composite numbers, I also recommend checking out 4th Grade Unit 1 of the IM curriculum for well-designed, ready-to-go lessons. (Link)

[UPDATE] Alyson Eaglen shared a great idea on Twitter. She said that instead of using cards with pre-printed target areas, she suggests rolling three 9-sided die and the sum is the target area. What a great way to bring in some bonus addition practice! If you don’t have 9-sided dice, you could always use five 6-sided dice or whatever combination of dice yields the range of target areas you’re interested in for the game. If you don’t have physical dice handy, Polypad’s free virtual manipulatives (Link) include a variety of dice under the Probability and Statistics menu.

# Our Venn Diagrams are One Circle

This past week my work life and my daughter’s school life came crashing together in the most wonderful way.

## I.

On the way home from school on Thursday, she asked if we could practice “take away.” At first we practiced numerical problems like “What is 3 take away 1?” and “What is 5 take away 2?” Eventually I asked her if I could tell my problems in a story. The rest of the ride home we told “take away” stories. I told a few, and then she wanted it to be her turn:

• “This one is sad. There were 2 cats and 1 of them died.”
• “There were 6 oranges on the counter. A girl ate 2 of them and they died in her mouth.”
• “There were 8 trees, and 3 of them got cut down.”
• “There were 6 roads, and 2 of them fell down.” (I was able to figure out she was referring to overpasses because that’s what we were driving under at the time.)

Slightly morbid, but she’s 6 years old, so I roll with it, especially since she isn’t usually this chatty about anything related to school.

Anyway, as we were getting closer to home, I remembered that the math unit she’s currently in in school uses some numberless word problems, so I asked, “Have you ever had a problem about some geese and some of them stop to rest?”

(Stunned silence)

“How did you know that?!”

“What about a problem about a boy who checks out some books from the library and returns only some of them?”

(Stunned silence)

“Yes! How did you know that one!”

“Because I wrote them.”

“What do you mean?!”

“I’m the author of the take away stories you’ve been working on in math class.”

And thus our two worlds – my work and her school – came crashing together for the first time ever.

I’ve mentioned to her before that I work with and help teachers, but it’s always been in the abstract. Finding out that I was the author of specific problems she’s encountered in her classroom just blew her mind. She wanted to see some of them when she got home. Knowing she probably won’t always be this interested in my work, I was only too happy to oblige.

## II.

As I was scrolling through the suggested unit plan to find the numberless word problems, I asked her about other tasks in the unit to see which ones she remembered. I asked about Bag-O-Chips, a 3 Act Task from Graham Fletcher, which was planned for the day after the numberless word problems, but she said she’d never seen it before. I have no idea how closely her teacher follows the unit plan, but lo and behold, the next day in the car when I asked what she did at school she said, “We did the bags of chips!”

We talked a little bit about the task in the car, and a little later as we finished up dinner I showed her the Act 1 video. Her eyes lit up. “That’s the video!”

We kept going back and forth between the image of what came in the bag and the image of what should have come in the bag. She happily used her fingers to figure out how many missing bags there were of each flavor.

I thoroughly enjoyed talking through the task with her, and what a pleasant surprise when she wanted to do another.

## III.

I’m not one to pass up an opportunity talk about math with my daughter, so I quickly scanned Graham’s list of 3 Act Tasks to find one I know we didn’t include in our suggested unit plans. I settled on Peas in a Pod.

Source: https://gfletchy.com/peas-in-a-pod/

First, we watched the video and estimated how many peas would be in each of the pods.

“I think there are 3 in this one, 4 in this one, and 10 in this one. No, 13 in this one.” (She estimated from right to left in case you’re wondering.)

“Hmm,” I said, “I think 3 is a good guess for the first one. I think there might be 4 or 5 in the second one, and I’m going to agree with your first guess of 10 for the third one.”

Estimation is a new skill for Kindergarten students. I talk about guessing and she talks about being right. She thinks the goal is to be the person who guesses the correct (exact) amount. I’m going to keep talking about being close and reasonable because over time I know her understanding of what estimation is will develop and refine.

Then we watched the reveal video.

Source: https://gfletchy.com/peas-in-a-pod/

“I wasn’t right and you weren’t right!” She exclaimed.

“That’s okay. All of our guesses were pretty close, even though none of them matched the exact number of peas. I was surprised that this one only had 2 peas in it. I thought for sure there were more in there.”

“Me, too.”

“Hmm, I have another question for you. How many peas are there altogether?”

“Let me count.”

“I want to see if you can do it without counting on the picture. How many peas were in each pod?”

“8 and 7…and 2.”

“So how could you figure out the total?”

At first she tried using her fingers. She counted out 8 fingers, and then continued counting from there. I couldn’t really tell what she was doing, but at one point, after lots of ups and downs of fingers, she said, “18.”

Pretty close!

I didn’t say that though. Instead I said, “Hmm, I wonder if that’s the right amount. What other tool could we use to check your answer?”

She decided to get her Math Rack to check, and as a complete surprise to me she said, “Can you make a video of me?” Make a video of you solving a math problem? Why, of course!

Watching her first attempt, it was fascinating seeing her trying to keep track of two separate counts: (1) counting on from 8, “…9, 10, 11, 12, 13, 14,…” and (2) counting the 7 she was combining with the 8, “1, 2, 3, 4, 5,…”

It seems like she abandoned the double counting  when she was so close to being done. I wonder if she sort of gave up and just continued counting to 18 since that’s what she had thought the answer was before.

I had a split second to think about how to respond. I didn’t want to confirm whether the answer was correct, and I wanted to see if she would be willing to try combining the three quantities again.

There was definitely a lot more accuracy when she separately modeled each quantity! I was impressed with the double counting she was attempting earlier, but in the end she was more successful when she could show each quantity separately and then count all.

It was a proud dad moment when she didn’t just accept 17 as the correct answer. She decided we should look at the picture of all the open pea pods to check. And, sure enough, when I held up the phone with the image of all the open pea pods, she was able to count all and verify that there were in fact 17 peas.

All in all, I’m over the moon. All year long I’ve asked her about school (and math), but up until now her answers have been fairly vague. (“I’m so surprised,” said no parent ever.) The most I’d gotten out of her before was that they did Counting Collections.

But now we’ve actually had a full blown conversation about the work she’s been doing in school, specifically activities I wrote or helped plan for our Kindergarten units. I’ve always loved talking about counting and shapes and patterns with my daughter since before she ever started school, but to have our worlds collide like this was really special. I enjoyed getting to share and talk about my work with a very different, and more personal, audience than I’m used to.

# A Dreambox Deferred

A year ago Tracy Zager wrote a must-read post called “My Criteria for Fact-Based Apps.” In it, she lays out her three non-negotiables for mathematics-related edtech programs:

• No time pressure
• Conceptual basis for the operations
• Mistakes must be handled properly

Tracy goes on to share two apps she does recommend, one of which is DreamBox Learning. After Tracy’s enthusiastic review, I wanted to get my hands on it and try it out. I invited a sales rep to our district for a demo, but that was underwhelming as always. For whatever reason, edtech companies tend to reveal only the briefest of glimpses of the actual student experience of their products. This is so frustrating to me!

Back in the spring I was part of a request for proposal (RFP) process that looked at various computer-based math programs. One of my biggest questions when reviewing programs is always, “What is it like for a kid using this program?” The reps will show you a few screens, but generally not enough to get a sense of what kids are really experiencing. Rather, the bulk of the time is spent talking about things like adaptive pre-assessments, teacher dashboards, and the plethora of reports that can be generated and dissected. Since the companies aren’t marketing to children, they focus their time and energy on the features that the adults will use. However, students are the ones using these products the most to (hopefully!) learn more about math. Their user experience is the one I care most about understanding and evaluating.

I did get some sample DreamBox accounts to play with, but I really wanted to see how it works in the hands of a kid, especially considering the adaptive nature of the program.

Enter my daughter, @SplashSpeaks. We’ll call her Splash for short. Splash is going to be 5 years old in March. She’s on the young side to be using the program – it says it’s designed for grades K-8 – but we have been doing so much counting and talking about numbers in our day-to-day lives that I thought it would be worth giving it a shot. Since the program is adaptive, I figured it would ensure she was in appropriate content.

Over winter break, I decided to create a personal account and start a two-week free trial. This post is about how, at least for now, I’m not going to subscribe now that the trial is over.

When Splash and I sat down at my iPad Mini to play DreamBox for the first time, she was excited to try a new app. The first few activities were a piece of cake for her. All they asked her to do was determine either “Which has more?” or “Which has less?” from two images of dots. The only challenge was paying attention enough to know which was being asked for. Her default was to assume that it was going to ask her to find the one with more. All in all, she did well enough and new activities started opening up for her.

I will say I’m impressed with the variety of representations she encountered in DreamBox. These included ten frames, dot images, math racks, and number tracks. It was interesting to see which ones resonated more with her. The math rack is definitely her favorite!

Sometimes the interactivity to complete one screen was a bit cumbersome for her. In the above example, she had to count the beads in the static image, create a representation of the same number of beads in the interactive math rack, count the number of beads again to make sure she remembered the number, count along the number track until she found the number she was looking for, click it, and then click the green arrow to indicate she’s done.

Whew!

Thankfully not every screen was this involved, but when they were, she would often skip a step. For example, she would build the number on the math rack and then jump down to the green arrow, forgetting to also select the number on the number track.

The first red flag for me that this may not be a good choice for her was her reaction whenever her answers were checked by the system. If she got the answer right, she would turn to me and smile, but if she got it wrong, she had a physical reaction of frustration. Rather than knowing it for herself, she started putting her faith in the system to tell her whether she had counted correctly. I didn’t feel comfortable with that shift in authority. I want *her* to trust that she counted correctly or built the number correctly, not wait for a computer to tell her. And I didn’t like how that subtle shift so dramatically changed her reactions to being wrong.

I will recommend that if your children use DreamBox, young ones especially, you should sit with them. There are some activities that ask for things Splash didn’t understand at first. For example, after building some numbers with the math rack, it started asking her to do it in the fewest number of moves possible. She had no idea what that meant.

Perhaps I should have said nothing and let her fail at the task. Since the system is adaptive, it might have shifted her back to other activities. However, considering how quickly the system brought her to this point in the first place, my guess is that after another activity or two she would have been prompted with these same directions.

I opted to explain to her what the phrase meant and she was able to start doing it on her own with the math rack. It was definitely more confusing with the ten frame, but even then I started seeing her grab larger chunks of dots rather than just counting out one at a time.

Here’s where another red flag came up. If you make a mistake on a screen that asked for the fewest clicks possible and then correct your mistake, the system will chide you for not getting the answer in the fewest number of moves and make you do it again. For example, let’s say you were supposed to drag 7 beads on the math rack but you mis-click and drag 6.  If you drag all the beads back and then click 7, which is what my daughter did, your answer is still wrong because it counted all the clicks you made on that turn. It doesn’t matter that your last click was the efficient one.

This caused my daughter a lot of distress because she felt pressure to make sure she was completing the task perfectly, but the mix of her 5 year old hand-eye coordination and my small iPad Mini screen meant this happened somewhat frequently. She had a similar issue with the number track where she’d be counting and pointing at the numbers on the track to find the number she wanted and accidentally click one of the numbers she was counting. In certain activities there is no green check mark. If you click the number track that’s it; the system thinks that’s your answer. It was frustrating to watch her getting discouraged at being told her answer was wrong even though it was a user interface issue.

Her frustration reached a breaking point when DreamBox started introducing Quick Images activities. If you’re not familiar, an image is flashed for about 2 seconds and then covered. The user has to select the number of beads/dots that were in the image. This just blew Splash’s mind! She can identify 1, 2, and 3 on sight, but if it’s 4, 5, or greater, she relies on counting one by one. This activity made her so annoyed the first time she did it. That is, until she had an idea. She hopped up and said, “I’ll be right back!” She came back with her personal math rack:

Suddenly the activity became much more do-able for her. By building the images herself, she started to notice that some images only had red beads and others had red or white. If the image had red or white then she learned she only had to count the white beads. Clever girl! She still hasn’t had the “a-ha” moment that all of the red beads are 5, but it’ll happen at some point down the road. I’m not worried.

Bringing in a math tool was a lifesaver for her. She had a renewed interest in the program and felt empowered using her tool to support her thinking. That is until she started getting Quick Images with dot images. This is where I’m curious how DreamBox gauges student ability with regards to numbers to 10. I already know my daughter is super comfortable with 1, 2, and 3. She clearly needs more work on 4 and 5. Numbers 6-10 I’m less concerned about though I know she can count them accurately.

The Quick Images activity is all over the map. It would show an image of 3 dots. Cool, no problem there. But then it would follow up with an image like this:

She took one look at this and was defeated. She had no idea how many dots there were. We haven’t played a lot of dice games yet, so she doesn’t know that arrangement of 5. And she doesn’t understand counting on yet so even though she can see two orange dots, that’s not useful for finding the total.

I let her take her best guess and get it wrong. I kept telling her it’s okay. If she gets it wrong the system knows she’s not ready for that problem and will give her a different one. This is where I ran into two big problems with DreamBox. First, the way it decides what numbers to give her seems random. After getting large quantities wrong, I figured it would adjust and only give her small quantities, but it kept ping ponging back and forth showing 3, then 9, then 2, then 8, then 10. In my head I was like, “Clearly she can’t figure out the big numbers, stop giving them to her!”

The other issue has to do with the length of the activities. Normally it seems like she answers 6-8 questions and the activity is over. There’s even a visual on the screen to help show progress. For example, a long dinosaur neck is inching along the bottom of the screen towards some leaves. In this same Quick Images activity, I saw that she was close to the point where the activity normally ends, so I encouraged her to do what felt like must be the last problem. And the one after that. And the one after that. And the one after that. It never ended! The dinosaur neck just kept inching and inching and inching toward that leaf. I felt like we were trapped in Zeno’s paradox. Each time, Splash got more and more upset and frustrated until she finally broke down in tears, and that’s when I ended it. If I had known the system was capable of extending an activity that long I would have backed out of it much, much sooner. As it is, I felt terrible! I love talking about and doing math with my daughter. The last thing I want is to bring her up to and well beyond the point of frustration.

We took a break from DreamBox for a day or two. When I asked her to try it again she said, “I don’t want to do it. I don’t like it.” That made me sad. I didn’t want to stop using DreamBox on the negative note of her last activity. I wanted to help remind her about all the amazing thinking she had been doing while using the program. I encouraged her to try again, but this time we would ignore those Quick Images activities. She was hesitant, but she agreed and we ended up having a good session. The next day we played again, but this time she chose an activity that I thought was something different but it turned out to be that dreaded Quick Images activity. Aargh!

Rather than give up, I took a quick look around the dining room and saw a tub of beads. Splash wanted to get right out of the activity, but I stopped her and said, “Why don’t you try using these to build the picture like you did with your math rack?” Building with beads sounded fun so she agreed. I also prompted her to look for small groups of dots in the pictures to help her. What a difference that made! She blew me away with her subitizing skills.

I was so proud of her! She managed to build every single image thrown at her. It wasn’t until the activity ended and said, “That’s okay, we’ll try Quick Images again another time,” that I realized the system was not as impressed with her performance. Apparently she was being timed. It took her a while to build and count each image. Even though she got every single answer correct, DreamBox considered it a failure and didn’t count the activity as complete.

A day or so later our 14-day trial ended and I was left with the decision about whether I should pay for a subscription. Splash clearly demonstrated some wonderful strategizing and thinking while using DreamBox, and I was tempted to see where it would take her, but I had a feeling in my gut that it wasn’t the right decision for her.

I couldn’t quite put it into words why until a week or so later when I read chapter 2 of Tracy Zager’s new book Becoming the Math Teacher You Wish You’d Had. The chapter is titled “What Do Mathematician’s Do?” In it, she shares the story of a primary classroom where students are asked what it means to do math. Initially their answers have to do with worksheets and giving answers. The teacher and Tracy work together to develop a mini-unit to open students’ eyes to what mathematicians really do. By the end of the unit, students are beginning to understand that math is about some wonderful verbs including noticing, wondering, asking, investigating, figuring, reasoning, connecting, and proving. They’re learning that math is all around them. Reading about the experiences of these students made me want to be in that classroom, experiencing that joy of discovery with them.

And then I thought of my daughter and all of the experiences we have daily with math. I realized that DreamBox might be better than nearly every other edtech program for practicing specific skills and working through a coherent progression of ideas, but it’s not the kind of math I want my 5 year old daughter to experience. I don’t want her worrying about whether a computer is telling her her answers are correct or whether she’s taking too long to come up with them or whether she’s finding them in the most efficient way possible.

In just two weeks I already saw that path leading to frustration and negative feelings toward mathematics. No thank you.

I want to continue down the joyful, meandering path we are already on where she investigates making shapes using her body and our tile floor:

Where she wonders about the biggest shape we can possibly make with plastic strips called Exploragons:

Where we figure out important things in our daily life, such as, “How many more days until the weekend?” and where we notice and play with math:

Down the road I might revisit DreamBox for my daughter, but not anytime soon. Lest you think I’m just being a harsh critic, I will still happily recommend it for parents and teachers who have older kids. When a child has more math under their belt and you want a system to be able to flexibly move backward and forward to meet their needs, then this is a great choice. It’s not perfect, but it’s far better than other programs I’ve seen. Kent Haines said it best:

But for a child just starting out and just beginning to develop her identity and relationship with mathematics, I’ll pass.

# We Can Make Shapes!

In my previous post I shared one of two mathematical conversations I had with my daughter this morning. Here’s the second.

I look over and she’s sitting cross-legged on the floor. It takes me a moment, but I realize she’s talking about the square tile she’s sitting on and the triangle she can see in the corner. Here’s a re-creation of it since I didn’t take any photos.

The third side of the triangle looked a lot cleaner with her crossed legs. This graphic of a child doesn’t quite work, but you get the idea.

“Oh! I see. How do you know it’s a triangle?”

As usual when I ask that question about a geometric shape – How do you know it’s a ___? –  she didn’t really say anything back. I turned around to put something in my lunchbox.

“Look! The triangle is smaller!”

I turned back around to look and she had scooted up on the tile. “So it is!”

With pure delight she exclaimed, “We can make shapes!”

She started scooting back on the tile and stopped when she got here.

“Is that a triangle, too?” I asked.

She looked down and thought for a moment. She slowly started scooting up until she got to the diagonal. Then she stopped and looked up at me.

She doesn’t yet know how to articulate what a triangle is, but she is clearly grappling with and making judgments about the “triangleness” of her shapes. It’s fascinating.

Even better, her exclamation, “We can make shapes!” makes me so happy. It’s such a simple statement, but it felt so empowered. She came to the realization all on her own as she moved her body back and forth on our tile floor.

# Counting Down to the Weekend

“Do I go to music class and swim class today?”

“No, today is Monday. Remember, I said you go to work for 5 days before you go to music class and swim class.” I hold up my fingers one by one as I call out, “Monday, Tuesday, Wednesday, Thursday, Friday.”

I put down all five fingers and continue, “So far we went to work on Monday and we’ll go today on Tuesday.” I put those two fingers back up as I talk.

Without skipping a beat she says, “Three more days! Today it will be 3, and then 2, and then 1.”

This was completely unexpected and so fascinating to hear! If only I hadn’t been in the middle of rushing to get dressed and ready to walk out the door to work. Looking back, I would have loved to ask, “How did you know there are three days left?”

In thinking about this conversation throughout the day, I’ve thought about all the play we’ve done with counting over the past several months. Fingers are a favorite of mine since they’re always close at hand.

In the car, one of the games we’ll play is that I hold up some number of fingers at my chest and ask, “Guess how many fingers I’m holding up.” She makes a guess and then I hold them up so she can see if she got it right. Nothing fancy, but it gives her a lot of opportunities to count and see quantities from 1 to 5.

Another game I like to play is, “Do you want me to show you 5 really fast?” She says, “Yes.” I put my hand behind my back and say, “Ready, set, go!” And then I whip out my hand with all my fingers out. She counts my fingers every time to prove there are 5 fingers, but I’m beginning to wonder if the counting is really necessary.

So I’m curious about how she knew it was 3 days until Saturday. The way I held my hand, she couldn’t see the three fingers that were down. Did she see them in her mind? Did she subitize? Did she count one by one super fast? There was hardly a heartbeat between what I said and her response. The counting back from 3 was really fast also.

Things to explore as we talk more.

I love being a parent and getting to have these kinds of conversations with my daughter. When she surprises me with a new understanding or insight, it’s like a wonderful gift. I treasure each and every one.

(Side note: Her Montessori school calls their learning time “work periods” so we’ve been calling it “going to work” since she started there a year ago. She likes the idea that she goes to work everyday like Daddy and Papa do. If I accidentally say something about going to school she’ll usually correct me, “No, I go to work!”)

[UPDATE 10/5/2016] This morning she asked a question she asks pretty much everyday without fail, “Is today a work day?”

“What did I say when you asked me last night?”

“It is a work day.”

I go back to eating my breakfast.

“We went on this day and this day, and this is today.” I look over and she’s holding up three fingers in front of her face. She’s grabbing the tip of her middle finger as she’s saying that this is today. She tells herself, “There’s two days left!”

Clearly our conversation yesterday wasn’t a fluke! She wasn’t even talking to me at the end. She was talking it out and making the observation all to herself. How cool!

A little later she’s in the kitchen and I ask her, “Can you show me how many workdays we’ve had on the Math Rack?” (By the way, we’ve had fun counting on the Math Rack, but I’ve never asked her to do anything like this before.)

She pulls over three beads, “One, two, three.” Then she holds up her thumb, touches it to the first bead and says, “One.” She holds up her pointer finger, touches it to the second bead and says, “Two.” Finally she holds up her middle finger, touches it to the third bead and says, “Three.”

“Can you show me how many days we have left down here?” I point to the bottom Math Rack.

She pulls over two beads, “One, two.” Then she puts her thumb, pointer, and middle fingers back up and moves her hand over to the two beads she just pulled over so that the two fingers that are still down are touching them.

I feel like she’s turned a corner developmentally and a whole new landscape has opened up. I’m so excited to explore it with her!

# Two Cats and Two Tortoise

Yesterday my wonderful co-workers threw us an adoption shower, and thanks to them our daughter is an even richer girl if you measure wealth in books.

One of the gifts was from Mary Beth. She said it’s a favorite thing to do with a favorite counting book, Rooster’s Off to See the World by Eric Carle.

Included with the book was a baggie that contained a small rectangular board and a bunch of small cards with animal pictures. My daughter pulled out the baggie and asked, “What’s this?” I said it was something we could look at while we read the book. My daughter didn’t want to wait to read the book, so this morning while I made breakfast, she plopped down on the kitchen floor to explore the baggie of cards on her own.

Within a few minutes I heard her say, “There’s two cats and two tortoise!” I looked over to see that she had filled her card with animal pictures. And sure enough, the card had two pictures of cats and two pictures of tortoises (or turtles, I’m not sure which yet). I like that all of the counting we do throughout our day has led her to notice and count things on her own without any prompting from me.

After she was sure I had seen the pictures, she cleared off the card and said, “I want to do more.” She put all the cards back in the baggie and started filling the rectangular board again. One thing that was really interesting to me was how she naturally made two rows of three pictures on her card.

When she finished filling the board up a second time I asked, “What do you notice?” I was curious if she would count again or if she would tell me something different. Her response was, “One chicken, one cat, and two fish.” A few seconds later she exclaimed, “And two frogs!”

Then she cleared the board again and started filling it with new animal cards. This time she chose only cards with fish on them. Her observation at this point was, “There’s a lot of fish on here!”

She started digging through the baggie for a few seconds before saying, “I need one more fish.” We haven’t really talked about “one more” very much so it was so interesting to hear her say that. Since she kept laying out the cards in the same arrangement, I’m assuming she could tell there was room for just one more card.

Picking randomly from the baggie wasn’t working so she pulled all the cards out and spread them out on the floor. She couldn’t find a fish so she changed the activity. She asked me, “Where’s the tortoise? Touch it.”

I dutifully touched the tortoise, and that was the end of the activity. She put all the cards back in the baggie and moved on to looking at another one of the books she received yesterday.

The one thing that came to mind as I watched all this unfold was Christopher Danielson’s message: Let the children play:

We adults have a responsibility to let the children play. We can be there to listen to their ideas as they do. We can play in parallel by getting our own egg cartons out and filling these cartons with our own ideas.

But when we tell kids to “make a pattern” or “use the colors”, we are asking the children to fill that carton with our ideas, rather than allowing them to explore their own.

I could easily have overtaken the activity by asking questions such as:

• How many tortoises are on the board?
• Are there more cats or fish in the baggie?
• How many animal cards are not on the board?

All of these are great math questions, but they’re MY great questions, not my daughter’s. I want her to develop her own questions and curiosities to explore. In the end, it was much more fascinating and rewarding for me to see the ways she came up with to explore the cards and to share the things she noticed about them.

Thank you, Mary Beth, for the wonderful gift!

# [tmwyk] Putting Away Blocks

We’ve had our foster daughter since November 2013. She came to us at 19 months old, and she is currently 2 years 5 months old. One of the things that I’ve enjoyed watching is how she’s become more strategic at putting away her blocks. They come in a box in a 6 by 5 array.

When she first started putting away the blocks after playing with them, she was, shall we say, haphazard about it. She would start by placing the blocks inside the box in any way she pleased.

I tried to recreate how she’d start since I don’t have any pictures of early attempts. This is probably even more neat than how she would have done it at ifrst.

This strategy worked fairly well until the box started to get a bit cramped. As she pushed blocks into smaller and smaller open spaces, some of the blocks would shift and move into the array configuration afforded by the box. Unfortunately some of the blocks would move in other directions, often ending up turned in ways that made fitting the rest of the blocks much more challenging.

This should give an idea of the troubles she would run into. When blocks were turned, they often made pushing new blocks into existing gaps much more challenging.

Despite the challenge this presented, I always marveled at her persistence in pushing and moving the blocks until she would get them all to fit nice and neatly in the box. Sadly, I don’t have any videos of her early attempts at putting away the blocks.

I did happen to take a video the other day showing how far she’s come from her early “trial and error” days. You’ll see there’s a lot more structure to her placement of the blocks. It seems informed by a mental image she’s developed of what the blocks should look like when she’s done.

Unfortunately she is not that verbal, so she can’t really talk to me about what she’s doing. Instead, I chose to sit back and watch. (So technically this is more Watch Your Kid Do Math instead of Talking Math With Your Kids.) Since this is clearly a task that she has always been able to figure out on her own, I’ve felt better keeping my mouth shut anyway. Clearly she’s been doing a lot of meaning making on her own over the past few months.

In some ways it’s excruciating to watch someone take over 3 minutes to put 30 blocks away, but as a parent and educator, I can’t help but be fascinated and wonder how she’ll be putting them away a few months from now.