We recently got our hands on a capsule from the Public Math math vending machine. Inside were four tiny tessellating narwhals. As soon as I showed Luke how they fit together, he started wondering about other ways to arrange them.

I told Luke that I didn’t know of any other ways to tile the narwhals and he insisted that I look the question up online. My Google searches were fruitless, so I suggested to Luke that we write our own blog post. He could look for new arrangements and whenever he found one, I’d take a picture. Then we’d publish our findings on the internet.

He was hooked by this idea and started calling me over every few minutes to document his discoveries.

“I found a new way to set it up! Call it Horn-to-Horn on All Those Horns.”

“Mommy! I found another way to connect the narwhals. Call this Rope.”

“There’s another way!”

“I just found another way!”

“I just made another thing with the narwhals, called Dragon Head Roar!”

When I asked Luke to make me a Zero Monster problem, I assumed it would be similar to the ones I had just been showing him. I was expecting to have to add and subtract various numbers of caps and cups. Instead, he showed me the below paper, which had a 0, 1 and 8. The 0 was being eaten by the Zero Monster, which he depicted as a circular mouth full of zig-zag teeth. There were no caps or cups in sight.

It took a few rounds of questions and explanations before I was able to figure out what he had drawn and what I was supposed to do with it. Barely visible on the page were the outlines of eight empty boxes from a worksheet on the reverse side. Luke had drawn numbers in three of the boxes and my objective was to identify which of the remaining boxes contained zeros. To guess a box, I had to draw a loop around it, like the loop he had drawn around the box with the zero.

My first guess was a big success because the box I chose had five zeros in it! My second guess was not so good, as the box turned out to have a one in it, not a zero. My third guess was the worst: it revealed the word “End” and that was the end of the game.

Playing this game gives me a lot of empathy for what it can be like to be a kid learning math from an adult. When Luke first showed me his paper, I had no idea what the drawings meant or what the goal was supposed to be.

I can see how Luke might feel the same way when I am showing him something new. Just because it makes sense to me doesn’t mean it will make sense to him, especially not right away. Unfamiliar notation in particular can be an intimidating barrier to understanding. (Trying to teach myself the lambda calculus via a “short and painless introduction” gave me a very visceral experience of this barrier.)

I wanted to win Luke’s game, once I understood the objective, but it felt like my success was completely in his hands. He was the one who knew the right answers; I just had to guess and see what happened. From my perspective, the game seemed completely arbitrary. Was there any internal logic to it? Were there pre-determined answers or was Luke making it up as he went along? I still don’t know!

Does Luke feel the same way when I challenge him to solve a problem that he’s never seen before? If Luke is not able to see the underlying logic in what I’m doing with him, it will feel just as arbitrary to him as if there were no logic.

Luke’s game reminded me that it’s not much fun to make choices when the outcome feels unpredictable. I don’t like making guesses at random that mostly turn out to be wrong, especially when the results don’t give me anything new to go on when making my next choice. It’s much more fun to feel like I know what I’m doing, or at least that I’m making progress in figuring something out.

My conclusion from all this is that I want to be careful not to go too far beyond Luke’s current level of understanding. I want to remember that some concepts and symbols are so familiar to me that they seem obvious, but they can be completely novel and perplexing to Luke.

Luke has a lot going for him: he has a strong natural interest in math, he picks up on new ideas quickly, and he’s got plenty of confidence in his own ideas and in his ability to figure things out. We also have a strong foundation of trust in our relationship, both in general and specifically with math. All this combines to make it pretty easy for me to course-correct when I accidentally overshoot what he’s ready for.

It can be a lot harder with my foster kids, who don’t necessarily have these advantages and who definitely don’t have that established relationship of trust. I have to step a lot more carefully to keep them engaged and avoid triggering frustration. This added challenge makes it tempting for me to do less with them, because the teaching does not feel as easy and natural as it does with Luke. But I know that the my efforts will bear fruit, both for the kids and for myself as I gain experience and skill.

As soon as I saw the illustration for Henri Picciotto’s Zero Monster lesson, I took a screenshot because I knew I had to do it with Luke. (Henri’s discussion of operation sense and symbol sense was also very thought-provoking. How can I support my kids in developing fluency with manipulating symbols in a way that builds their understanding of the underlying concepts rather than replacing it with mindless rule-following?)

This morning during breakfast, I invited Luke to do an activity with me about the Zero Monster. He immediately started asking questions and speculating about the Zero Monster and I realized I had not actually planned out any details. Fortunately I had some time to think while I toasted bagels and set the table. As I came up with ideas, I threw out little tidbits of a story.

“Well you see, the Zero Monster likes to eat zeros.”

I wanted to make it clear from the outset that the cup and cap symbols represent two different things, and that you can’t just turn a cup upside-down to make it into a cap. I had the idea to make the story about literal cups and caps.

“One time the Zero Monster went to the park. At the park, he saw a big mess. Some of the kids at the park had been littering!”

Luke started telling me about a time that he saw a spoon and some vanilla ice cream on the ground at the park. This was helpful because it gave me some more time to think!

Once breakfast was on the table, I sat down with some paper and a couple pencils. First I drew a picture of the Zero Monster. Luke watched with fascination.

Me: Pick a number between one and ten.

Luke: Five.

Me: Okay, so when the Zero Monster was picking up litter at the park, he found five caps, like this. Now pick another number.

Luke: Eight!

Me: And he also found eight cups.

The first cups I drew were very shallow. Luke urged me to make the sides taller, which was a good move. It made them more cup-like and they did not look so much like the caps.

When I got up from the table to get Nathan some cereal, Luke showed me how he had added horizontal lines to my caps to turn them into baseball caps. I acknowledged this and continued on with the story.

Me: And this is what the Zero Monster did. He put caps on the cups, and they turned into zeros! And he ate them. What did he have left over, after he ate all the zeros?

Luke: Three cups. And there was a giant cup that was a trash can, and he put the extra cups in the trash. It had room for four cups.

Luke illustrated the trash can while I wrote a new problem on another paper. Nathan wanted to draw t’s so I got him his own piece of paper and some markers. He got himself a wipe and was disappointed when he couldn’t erase the marker with it. I explained that unlike the dry erase boards we had used the other day for Exploding Dots, it was the wrong kind of marker and the wrong kind of paper to erase with a wipe.

I showed Luke my new question:

2 cups + 7 caps + 3 cups + 1 cap. How many are left over?

For this one, Luke told me his thinking and I wrote it down in an organized way. Two cups and three cups is five cups. And eight caps (because he found seven caps and one more cap).

Then I drew the picture and we saw that there were three caps left over after the Zero Monster ate the five zeros.

Next I had the idea to use symbols instead of words to represent the cups and caps. This was faster for me and I hoped it would make it easier for Luke to do his own writing.

4∪ + 1∪ + 3◠ + 2∪ + 2◠

For this one, Luke noticed that the two cups and two caps at the end combine to make two zeros. I rewrote the expression without the last two terms and we agreed that this would be equal to the original expression.

4∪ + 1∪ + 3◠

He drew a picture of another trash can with two caps going in the top and two cups at the bottom, and declared the answer to be two. “Two caps or two cups?” I asked. He wasn’t sure, so I encouraged him to think through it again.

To find the final answer, I’m pretty sure his thought process was to put one cap on the single cup and the rest of the caps on two of the four cups, with the understanding that there were two cups left of the four that did not get caps.

He explained that the Zero Monster was holding the two cups upside-down when he put them into the trash can, and they landed right-side-up at the bottom.

I gave him the pencil so he could write his answer. First he just wrote a big two, so I encouraged him to add the symbol and he added the cup shape.

I like the role that units play in this activity. Most of the math Luke has done so far has either had no units, or the units have been decorative. We can’t ignore the units in these Zero Monster problems because they actually mean something!

Next I made an equation with an unknown term: 5∪ + ☐ = 2∪

Luke’s first move was to tell me that we had to use a minus instead of a plus. When I insisted that it was supposed to be a plus and that I hadn’t made a mistake, he told me the problem was impossible.

I forget what my response was to this. I think I encouraged him to try putting something in the box to see what would happen. He wrote ◠1 and I drew the picture to show how we’d get one zero for the Zero Monster to eat, and there would be four cups left over. But we want to have two cups left over, so it matches the 2◠ on the other side of the equation.

From there, Luke was able to see that we needed three caps, so he changed his answer to ◠3.

I’ve been reading a lot of Dan Meyer lately (I’ve made it up to 2012 in his blog archives) and I really like what he says about letting reality be the authority, so that students do not need the teacher (or textbook) to verify their answers. I guess that’s one big benefit of drawing pictures or using manipulatives — it allows us to convert abstract symbols into something more concrete so that Luke can evaluate for himself whether his answer is correct. We can move up and down the ladder of abstraction together, switching back and forth as-needed until he’s formed his own connections of understanding with the symbolic representation of these concepts.

Next I told Luke it was his turn to make a Zero Monster problem for me. This turned out to be an interesting detour, because the problem he came up with was very different from what I had expected. We also explored some different ways of drawing the Zero Monster, before I brought our attention back to caps and cups by showing him the following problem:

9◠ + 5∪ + ☐ = 1∪

Luke’s first step was to match up five of the nine caps with the five cups to make five zeros. He explained that there would be four caps left over and I showed him how to write the simplified equation:

4◠ + ☐ = 1∪

He was stuck on this one for a while, but after a hint or two from me, he decided to try adding “a lot of cups.” I prompted him to pick a number and he chose eight. I did not have a chance to finish drawing the four caps and eight cups before he revised his answer to five cups. We drew a picture to confirm, and I pointed out that both sides of the equation were the same because when you add four caps to five cups, you end up with one cup after the Zero Monster eats all the zeros. 1∪ = 1∪

Then I asked Luke if we could put the same 5∪ answer into the original equation. He thought that would work, so we tried it out and drew the picture. Sure enough, we had one cup left over! Luke was very satisfied to have figured out such a tricky problem.

He was excited to do more, so I gave him another one:

2◠ = 3◠+ ☐ + 7∪

At first he read it as two caps plus three caps, so I pointed out the equals sign. He was surprised that it went “the other way” and asked why. I explained that it doesn’t matter which side the equals sign is on in an equation.

Luke asked me to draw another Zero Monster in the style he had come up with. Then he drew a Zero Pet with leashes to connect it to my Zero Monsters.

His first guess was 2. I think he was still thinking in terms of 2 + 3 + ☐ = 7. I asked him whether he meant two cups or two caps and he reconsidered. Then he tried 1∪ and I drew the picture to show what that would look like. He could see that it didn’t work, but was confused about what to do next until I redrew the picture to show what we already knew: two caps on one side, and three caps with seven cups on the other.

He announced that we needed nine caps and wrote 9◠ so I drew nine additional caps. He clarified that he meant nine caps total, including the three that we started with. I asked him how many we needed to add and he said six, revising his answer to 6◠. I drew the six extra caps and he had me connect them to the cups to make seven nice zeros which we fed to the Zero Monsters. I wrote 2◠ = 2◠ at the bottom to show that both sides of our equation were the same.

Again, Luke was very pleased with himself that he had figured out such a tricky problem. He drew me a Zero Cat and we played a few celebratory rounds of tic-tac-toe.

Now I’m thinking about what I want to do for next time. I definitely want to decide on the problems ahead of time instead of writing them on the fly. My tendency is to make them more complicated because I assume the simple ones will be too easy, forgetting that what seems simple to me is not necessarily simple to Luke. I’m thinking I’ll try making a page or two of the simplest ones to see if he can do them on his own, without hints from me. Will he find them boring? I guess I’ll find out!

With this first round of problems, I was very hands-off in terms of strategy. I encouraged him to try things and see what happened, and I drew pictures to illustrate his guesses, but I did not suggest any particular methods or ways to think about the problems.

I’m thinking next time I will be explicit about showing him the strategy of isolating the unknown by adding the same amount of cups or caps to both sides of the equation. I can prep by making some medium-level equations that are amenable to that strategy. Anything that has caps on one side and cups on the other seems like it will be good for this, since those were tricky for him to figure out in his head.

I want to continue showing him examples of equations that don’t fit the standard ☐ + ☐ = ☐ pattern. Maybe I will show him two or three different versions of the same equation (i.e. the same terms but in a different order) and ask him if he can show me any more ways to write it. The extra complexity of the cups and caps is not relevant here, so instead of incorporating the Zero Monster, I’ll show him Denise Gaskin’s wonderful Substitution Game. I think we’ll have fun seeing how long of an equation we can make together.

Luke: If twenty-five can make a 5×5 square, can it also make a 4×6 rectangle? Because five-and-five, and four-and-six?

Me: How would you figure that out?

Luke: We could draw a 5×5 square and a 4×6 rectangle.

Me: Ok, I’ll get you some graph paper.

Luke: Actually I only have to draw a 4×6 rectangle.

He drew the rectangle and counted the sides to make sure they were correct. Then he counted each square in the middle. I’m not sure if he made the dots on purpose or if they just happened that way as he used his pencil to keep track of which square he was on.

When he was done, he announced the result: “It’s only twenty-four! It’s one less. Twenty-five is one more!”

I was pleased by our little exchange but did not think to take it any farther. In the past couple of days, however, I’ve read through a big chunk of Christopher Danielson’s blog Talking Math with Your Kids, which has inspired me to start asking more follow-up questions that can extend Luke’s thinking a bit beyond where he might end up on his own.

Luke: 3×5? Well I know it can make 2×8. Let’s find out. But we’ll also need some graph paper.

I went upstairs to get the graph paper and took the opportunity to type up our exchange so far. The extra minute of delay gave Luke some time to think.

Luke: Let’s bring that downstairs. But I already know. You’ll have to figure it out.

Me: If you already know, write your answer down in a secret place. Then I’ll figure it out and we can check whether we got the same answer.

I drew a 4×4 square and labeled it with a 16 in the middle. Then I drew a 3×5 rectangle. Luke told me to draw lines to connect each square of the rectangle with a square of the square. I suggested cutting out the shapes and putting them on top of each other but he insisted on the connecting lines.

We found that there was one square out of the 16 that did not have a partner in the 5×3 rectangle, which meant that it must only have 15 squares.

Luke ran to get the paper he’d hidden with his answer: 15

I got up to get Nathan some more toast and pondered my next move. I wanted to point out how both shapes have the same perimeter but we haven’t talked about perimeter yet and I wasn’t sure how to introduce the concept.

When Luke had first posed his question about twenty-five making a 4×6, he had hinted at his reasoning. I assume he was thinking that 4×6 would work since 4+6 = 5+5, though I didn’t stop to confirm that with him. But if that was the case, it would connect to the concept of the perimeters of each shape being the same.

The graph paper we were using was very small, so it was hard to point at individual line segments and talk about them. I’ll have to see if I can dig up anything bigger for the future — I think I have 1/2 inch gridded paper somewhere.

I decided to try drawing some snakes of different lengths and talking about how long they were, with the goal of having them form various closed shapes once we’d established the snake as its own entity, separate from the area it might enclose.

I drew a zig-zag with 10 segments and labeled it a 10-snake. Then I drew a tiny 1-snake.

Luke loved this idea so much that he took over my pencil and drew a very wiggly snake that kept going till it reached the edge of the page.

Me: Wow, that snake is so long! I bet it’s a 50-snake.

Luke: Watch this!

He went back to the beginning of the snake and extended it backward toward the other edge.

Luke: Now how big is it?

Me: I don’t know, but it’s got to be bigger than 50. Maybe it’s a 60-snake.

Luke: Or a 100-snake! These snakes are working hard.

We wrote down our guesses and then he started to count. It turned out to be an 81-snake!

Now I was ready to take it to the next level. I drew a little 1×2 rectangle and labeled it a 6-snake.

Me: I made a six snake.

Luke: Are you sure that’s not a two snake? Or a four snake?

Me: Yeah, because it goes like this, see? One, two, three, four, five, six. And then it ends up back at the beginning.

Luke: Look at my one snake.

He took the pencil and drew a long straight line across the page. Then he labeled it as a 1-snake.

I took a moment to consider my next move and then cried out in mock outrage, “No, no, no! That’s too long to be a one snake!”

I drew and labeled a couple of 1-snakes and then a straight-lined 2-snake, to show him that a 2-snake was bigger than a 1-snake.

Luke told me the long edges of my rectangle counted as one and drew a true 6-snake underneath my false one.

Then he started writing a declaration. He abbreviated it to save on writing but dictated to me what it meant as he went along. Then I wrote it out in full underneath so we wouldn’t forget what it meant:

1LLcaaL1S

1 Long Line counts as a Long 1 Snake

At this point I decided to concede, especially since I could see how my initial examples had been ambiguous. I flipped the paper over and drew some examples of 1-snakes and 4-snakes using Luke’s definition, where each straight line segment counts as one no matter how long or short it is.

I think it’s really cool how Luke argued so confidently for his terminology! We had been talking about polygons earlier in the morning, so it makes sense that his instincts were to place more importance on the number of line segments than on their length.

I’m thinking that if I want to get him thinking about lengths and perimeters, I’ll need to get creative. Maybe I’ll get out the tape measure and leave it lying around somewhere. The train tracks in the playroom seem like a promising spot….

Now that our foster son Noah has moved out, Luke has been complaining a lot about being bored. I’ve been on the lookout for activities we can do together that hit the sweet spot of being educational enough to feel satisfying for me while not being “too hard” or “too boring” for him. I also want to help him fill out his repertoire of fun activities to do on his own, since I’m not able to spend all day long giving him my undivided attention.

The other day while he was hanging around my desk waiting for me to be done with work, he had the idea to make a pickup truck out of paper. Although I was worried that his ambition would outpace his skill level and he’d end up frustrated and crying, I certainly wasn’t going to discourage him from trying. I helped him collect paper, tape and scissors, and he went to work.

The first difficulty he encountered was in cutting circles for the wheels. He was trying to draw circles free-hand and it wasn’t working very well. I found a core from a used-up roll of tape and showed him how to trace around it to make his circles. He was very pleased by how well they turned out!

The next challenge was to cut the circles out. I gave Luke a few pointers on how to hold the scissors and paper (two thumbs up!) and reassured him that it would get easier with practice. He cut out a few circles but was struggling because he kept drawing them in the middle of the paper. I showed him how to trace the circle in the corner of the paper so he wouldn’t have so much extra paper hanging off and getting in his way while he cut.

Meanwhile Nathan sat on my lap and did some cutting of his own while I drafted invoices and sent out purchase orders. He was delighted when I traced lines so he could cut squares off the corners of his paper.

By the time Luke was finishing up his last circle, he was already remarking on how much easier it was compared to when he first started.

After that he worked pretty independently, cutting out each piece and putting it in a pile: four wheels, two wheel bases, one big rectangular base for the whole truck, and various pieces of the cab including windows.

When it came time to do the taping, he was having trouble holding the cab pieces together in a three-dimensional arrangement while placing the tape, so I offered to hold the paper in place for him. He did the rest of the taping on his own and was very satisfied with the result.

I admired Luke’s truck and took a picture, but it wasn’t until later when I was reflecting back on the day that I realized what an accomplishment it was. He had planned out his design, eyeballed the measurements and figured out how to connect 2D shapes to make a 3D structure — all without any tears of frustration, and with very little input and assistance from me. I suppose he has quite a bit of block- and Lego-building experience to draw from, but still; I’m impressed by how fluidly he dove into this new medium.

Since then I’ve been thinking about what other concepts or techniques I can show him, in order to build on this foundation. One thing I realized is that he hadn’t used any folding in his construction: each face is a separate piece of paper connected to its neighbors by tape. I’m thinking I want to do an activity soon where we design nets that we can cut out and fold. We can start by making some characters from Numberblocks, since they are geometrically simple but will be fun to decorate and play with.

My other idea is to start doing more origami with him, since he did well when we made our origami books. Paula’s recommendation that we try her origami pockets has been on my mental to-do list ever since, but Luke’s pickup-truck bumped it up to the top. When he asked about something to do with me yesterday, I suggested we make origami pockets and we got some paper.

I walked Luke through the steps and we each made a pocket out of a sheet of printer paper. When he was done, he was excited to decorate it orange with his orange machine and fill it up with stuff. Even his orange machine fits in there!

After that came the fiddling and with the fiddling came the questions.

Looking at the flaps: “How do these stay closed instead of popping open?”

Unfolding the pocket all the way and observing all the undecorated areas: “How is this just this much coloring?”

Folding it back up: “Why is this flap flapping open?”

Randomly out of the blue: “I didn’t know these were pockets instead of pentagons.”

At some point, I posed a question of my own:

“If we had a bigger piece of paper we could make a bigger pocket. Do you want to try that sometime?”

“Yeah, let’s do it now.”

Off I went to the closet to search for some bigger pieces of paper. I found some wrapping paper and some kraft paper and we went downstairs so we could spread out on the dining room table.

After we folded a large kraft paper pocket, Luke cut out a tiny square and wanted to see if he could make a tiny pocket. I thought it might be hard to fold those little folds with such thick paper, but it worked really well. Luke searched around for something to put inside.

“A tiny pocket that can fit a bean. It can be a cannon that shoots out a bean.” It’s hard to aim, but he was able to get some good distance by squeezing it just right.

The wrapping paper had a helpful grid on the back so we counted out a sixteen-by-sixteen inch square and cut it out.

Unfortunately when we made the first diagonal fold, we discovered that the gridlines were not as helpful as we had hoped — it wasn’t quite square! But it wasn’t too hard to trim the extra half-inch off the longer side. I walked Luke through the steps and we worked together to fold up our big red snowflake pocket. It makes a festive hat!

Luke was not satisfied with the size of our 16×16 pocket. “Look at this one. It’s only this big. I thought it would be way bigger. Let’s make a bigger square, like twenty by twenty. Will that be a too-bigger pocket?”

While I was examining the wrapping paper to see how big of a square we could get, he cut a few Numberblocks characters including Twenty-Five and Sixteen. With a few extra cuts, he showed me how Sixteen is “A square… 🎵Made of squares!🎵”

We cut out a bigger square and Luke led the way in folding it, along with some more commentary:

“We have a twenty-two by twenty-two square. Is this pocket going to be bigger than the last pocket?”

“Is there any more steps except for up and this?”

“It’s way bigger than the other one. Look how big it is compared to the other one. It’s so big we can… make a double pocket!”

After that, he was unstoppable. He had the knowledge and he was going to use it to make more pockets. “Can we make pockets out of these little scraps of paper? Hmmm….”

“Is this a small activity? It’s a very tiny activity. Heh-heh! We did the activity! A teeny-tiny pocket that can just fit a bean, again.”

“I know how to make pockets myself. All these pockets are already decorated. I know just how to make these pockets.”

“A pocket that really holds onto the bean. The bean never drops.”

I was inspired by his four mini-pockets to make my own quadruple pocket.

If you pop open the bottoms of the pockets, it can even stand up!

“How many kinds of pockets can we make? Small pockets, big pockets, we can make tiny pockets, huge pockets, pocket pockets, how many kinds of pockets can we make? Large pockets, teeny pockets, all kinds of pockets.”

This story is a sequel to Pickle and the Boolean Logic Baby Gate. I told it to Luke and Nathan at bedtime, and Luke chimed in with his own additions that sometimes changed the course of the story from how I was intending it to go.

One day Pickle was looking in his closet to see if he could find his snow boots. He searched and he searched, and while he was searching, he found a function. He looked all over the function but he couldn’t find any label that would tell what the function did. It was a mystery function.

“I’ll put a banana in it,” said Pickle. So he put a banana into the function and out came a peeled banana and a banana peel.

“That’s interesting,” said Pickle. Then he put an orange into the function, and out came a peeled orange and an orange peel. Pickle tried putting in an apple, and out came a peeled apple and an apple peel.

“This function would be a very good function for the kitchen,” said Pickle. He put a label on it that said “Peeling Function” and he put it in the kitchen.

Then Pickle remembered that he still hadn’t found his snow boots, so he went back to the closet to look for them again. After a little while, he found another function. It also did not have a label.

“I wonder what this function does,” said Pickle. He put a red Duplo into the function, but nothing came out.

“Oh no!” said Pickle, “I don’t have my Duplo anymore. I should not put special things in this function. I should test it with something that’s not important, like a piece of trash.”

Pickle took the function outside to the trash can and found a big pizza box. He put the pizza box into the function and out came one hundred pizza boxes!

“No, no, no!” said Pickle. “That’s not what I wanted at all!”

“This is a bad function,” said Pickle, and he put the function into the trash can and went back into the house to keep looking for his snow boots.

After Pickle went inside, a squirrel found the function in the trash can. She took it out and put a nut in it, and out came one hundred nuts. The squirrel was very happy about this so she took the function back to her nest. Then she got to work burying her one hundred nuts so she could eat them later.

Meanwhile Pickle was looking all over the place in the closet, trying to find his snow boots. Instead, he found another function.

“What does this function do,” asked Pickle. “How should I test it?”

He didn’t want to test it by putting in a Duplo in case the Duplo went away forever, but he put a Duplo in it anyway and out came one hundred Duplos.

“Wow,” said Pickle. “This is an amazing function. I definitely want to keep this function, but I had better put it in a safe place. If Baby Pickle gets it, he might put in a dirty diaper.”

So Pickle put the function up on a high shelf and went back to the closet. As soon as Pickle was out of the room, Baby Pickle climbed up to the shelf and got the function. He put a dirty diaper into it and out came one hundred diapers — and the worst thing is, they were poopy diapers!

Baby Pickle put another dirty diaper into the function, and another one, and another one. He kept putting dirty diapers into the function and it made a pile of dirty diapers that got bigger and bigger.

“What’s that poopy smell?” wondered Pickle.

Baby Pickle got tired of smelling that poopy smell, so he went outside and left the poopy diapers in the playroom.

Then Daddy Pickle and Mommy Pickle came and cleaned up the diapers. They went outside to the backyard and brought down the function the squirrel had hidden in her nest. When they looked at it, they saw that it had a conditional statement on the side.

The conditional statement was: “If x equals trash, then output x times one hundred. Otherwise, output nothing.”

“That’s backwards,” said Mommy Pickle. “We need to switch it around.” So they switched the conditional statement around so that it said, “If x equals trash, then output nothing. Otherwise, output x times one hundred.”

Then they put all the dirty diapers into the function and nothing came out. But when the squirrel put nuts into the function, they still came out. They came out both ways because they were not very good objects, and not very bad objects.

Meanwhile, Pickle was still in the closet looking all over for his snow boots, but he was not finding them anywhere. All he found was his rainboots and then he found another mystery function.

Pickle put a ball into the function and out came a snowball. Then he put a tire into the function and out came a snow tire. Then he put bran flakes into the mystery function, and out came snowflakes.

“I know what to do,” said Pickle. He put his rainboots into the function and out came snow boots.

Inspired by Paula Beardell Krieg’s wonderful blog post, yesterday I made little origami booklets with the kids. Rather than making a big announcement, I decided to take the sneaky approach: after laying out all the supplies, I just sat down quietly at the table and started to fold. Nathan climbed up into my lap right away but the big kids were busy racing around the house gathering nuts for their squirrels.

After I had folded my first book, Nathan dictated the contents for me: “Big O! Key. Door unlock it. Key.” I drew his pictures and added labels next to them. Then we read it together a few times to his great delight.

Eventually Luke and Noah trickled over and wanted to make books of their own. I demonstrated the steps to each of them, making sure to keep my paper oriented the same way as theirs to keep it as clear as possible. I haven’t done much paper-folding with them before so I wasn’t sure how difficult they’d find it, but they both did really well!

After reading Paula’s story about making these little books with a class of first graders, I was excited to see what kind of creativity these blank pages would unlock in my kids. What would they draw? What would they write? I couldn’t wait to see it!

The joke was on me because as soon as Luke was done with his booklet, he picked up my finished book and started copying it picture-for-picture and word-for-word. By the time Luke was done, Noah had finished folding and was waiting for his turn to copy my book!

When I remarked that Luke had made a copy of my book, he pointed out a difference I hadn’t noticed: he had reversed the page order so that his book read from right-to-left.

Noah’s changes were more obvious: in addition to using multiple colors, he condensed my double-spread layouts to single pages by putting the text on the same page as the pictures. This left him with some blank pages to fill, so he added a polar bear, a lion, and a “The end” on the back cover.

I was surprised to see that Noah also wrote his book from right to left. Unlike Luke, however, Noah maintained my page order by starting with my last page and working backwards through my book.

I gave Nathan a book to write in and he filled a few pages with O’s.

When I mentioned to Luke that his pages would be in the same order as mine if we looked at his book through a mirror, it gave me an idea: “Luke, I’m going to write you a secret message!” I filled out another booklet with mirror writing. Luke was very eager to get his secret message but when I finally gave it to him, he was a little confused about what to make of it. I gave him a hint by telling him I had written it in mirror code and he rushed off to the bathroom mirror.

He was very excited to discover my secrets!

This morning, the kids were rereading their books and hiding them in mailboxes for each other. With this renewed interest, I brought out supplies for second round of bookmaking. This time Noah wanted to make a bigger book so I showed him how to fold his paper into quarters. This was not enough pages so we made another set and stapled them together.

While Noah was filling his book with pictures of animals, Luke asked to have input into the book I was making. I guided him through a simple story structure. Who are the characters? What is the problem? What happens first? The resulting story involves a tractor chase, a bear blocking the escape route and some good dodging reflexes.

Now they’re acting out their story in the kitchen, with Luke as the tractor, Noah as the bear and Nathan as the little kids running away.

What a fruitful activity this has been! It’s always fun to see where we end up when things go in a completely different direction from what I was expecting.

Now my problem is choosing what to show them next: Bigger books with more pages? Mail boxes and envelopes? The Mirror Game? More secret codes? I guess I’d better observe their play over the next day or so to see if anything sticks out as a particular interest.

As Luke and I have continued to play The Secret Number Game off and on, I’ve discovered just how versatile it is. Because it is so easy to change up the mathematical concepts involved, it’s become the kind of simple-but-high-leverage game that makes for a good instructional routine.

To keep Luke engaged, the clues I give him have to be hard enough to be interesting but not so hard that he gets frustrated. This forces me to think up new kinds of clues to give:

“If you have my secret number and you double it, you’ll get twelve.”

“If you have my secret number of cars, you will have twenty tires.”

“If you have twelve donuts and you give them to four people, my secret number is the number of donuts each person has.”

“My secret number is the cardinal number associated with the set of people living in our house.”

I also get feedback on his comfort level with a topic when I see how he handles a clue that I’ve given. Sometimes if he can’t figure out my clue, he’ll request a different clue that’s easier for him to work with:

Me: My secret number is, I’m going to call it X. Four plus X equals nine. Do you know what my secret number is? Luke: No. What does it equal plus twenty? Me: X plus twenty equals twenty-five. Luke: It must be five. Me: Ahh, you guessed my secret number! Ahhh! It is five. Luke: Make another one!

I aim for clues he can figure out in his head but if a clue turns out to be too tricky, I’ll bring out some manipulatives.

Me: x + 20 = 30. What’s my secret number? Luke: 30! Me: Let’s put that in for x. 30 + 20 = 30. Does that sound right? Luke: Yeah! Me: I think we need to get the ten rods for this one.

Words of wisdom from Denise Gaskins, my source for this game: “Don’t forget to take turns. If you let the kids make up problems for you, it becomes a game instead of busywork.”

Incidentally, this advice has given me more empathy for my kids because sometimes they want me to guess an answer without giving me any clues! I find this really frustrating, but then I realize that’s probably what it feels like for the kids when I’m trying to get them to figure something out on their own but it’s not clicking for them and they’re just stuck. Who wants to keep guessing at an arbitrary answer when you know you’re just going to keep getting it wrong?

The process of coming up with a secret number and a clue is its own kind of challenge for Luke. When he’s working backward from his secret number to figure out his clue, I am careful to practice selective listening so that I don’t hear him give away the answer.

Sometimes he’ll just throw out a clue without bothering to figure out his own secret number first.

Luke: My secret number is how many things our car has on it. Claudiu: What are things? Luke: My secret number is the number of things our car has inside. Claudiu: Our van? Do you know your secret number? Part of a secret number is you have to know what it is. If you don’t know it, it’s a mystery number that no-one knows.

It’s always interesting to see where this game takes us.

Luke: My new secret number is the number of dots on this die. Claudiu: Twenty-one. Luke: …14, 15, 16, 17… did I count on the three side? 1,2,3… 4,5… 6,7,8,9,10… Claudiu: Do you want to know a secret about dice? Luke: Yeah. Claudiu: The opposite sides add up to seven. Look, pick a side. Luke: Three. Claudiu: What’s the opposite? Luke: Four. Claudiu: What do they add up to? Luke: Seven!

This is a game I played with Luke a few years ago. I want to introduce it to Nathan soon since he’s been interested in counting lately.

I made a log with five green frogs and a pool of blue water. As I sang each verse of Five Little Speckled Frogs, I showed Luke how to have a frog jump off into the pool. Then we counted how many were left on the log to see what number we needed for the next verse.

Five little speckled frogs
Sitting on a speckled log
Eating a most delicious fly
Yum yum-yum yum yum yum
One jumped into the pool
Where it was nice and cool
Now there were four little speckled frogs.
Four little speckled frogs...

It wasn’t long before Luke was the one making the frogs jump and telling me how many frogs were left.

I’ve discovered another great math song in the vein of Baa Baa Black Sheep. Again, the big epiphany is that I don’t have to count down by ones. Starting at a higher number doesn’t mean I have to drone on forever if I let them fall out of bed by the dozen!

It’s like a musical version of Jenna Laib’s Skip Count Game! Though my preference for counting down by the same number fall out in each verse is generally shouted down by the peanut gallery.

There were one hundred in the bed
And the little one said,
"Roll over! Roll over!"
So they all rolled over
And ten fell out.
There were ninety in the bed
And the little one said,
"Roll over! Roll over!"
So they all rolled over
And twenty fell out.
There were seventy in the bed
And the little one said,
"Roll over! Roll over!"
So they all rolled over
And seventy fell out.
There were none in the bed
And the little one said,
"Ouch! I got hurt on pants!"

When we get tired of this one I can pull out another backwards counting song like Five Thousand Little Speckled Frogs or Ten Million Monkeys Jumping on the Bed.