Making Books with Money

April 27, 2017



Oh my gosh, working with second grade students is so rich.

They have skills, they are enthusiastic and uninhibited, and tapping into their learning curve is delightful.

Windshield, with George in the driver's seat

Windshield, with George in the driver’s seat

I’m working with three sections with about 22 students per class, so I’m getting to see about 66 different ways that students are making sense of the 100 cents project that I described in my last post. (oh, there’s an unintended pun in that last sentence, did you get it?

Abstract design

Abstract design

Short recap: students were given images of coins, which added up to $3.00, from which they chose $1.00, or 100 cents, worth of coins to create a design.

These students hadn’t started studying money yet, which was fine. Most students seemed to understand how much coins were worth, though certainly a few students had no idea about the value of coins.

It was fun, when adding up the value of nickels, to say, Now you know why it comes in handy to count by fives.



Making the wallet-book to house the 100 cent images, then making the images was what we got done on the first day. Separating out 100 cents was certainly the most challenging part of the project. The designs flowed freely.



Day 2 was a bit more challenging, but I think that the toughest part was just communicating to them what I was looking for, which was for the students to make matching arrays of the coins that they used in their designs, then providing the equation which showed that the value of the coins equal 100.



Turns out that this array-making uncovered a few mistakes. For instance the airplane pictures above was five cents short, so he added a nickel on to the bottom and all was well.

Person in landscape

Person in landscape

There was a wide range of simplicity to complication of images.



If students didn’t have enough coins of a certain value left from their original 300 cent to making the matching array, they would exchange change with another student, at least that was the plan, which worked fairly well. I did bring lots of extra coins, for moments when it seemed better just to hand students what they needed.

Still, everyone should have had 100 cents left over. These coins got glued on to a pocket of their wallet book, along with a statement of the value of these coins.  That little black folder that contains the 100 cent image now has an enlarged section of a colorful buck glued on to the front. After all that figuring and adding, it was great to end yesterday’s class with some playful coloring in.

Okay, one more day with these students. The next piece that goes into the wallet-book has to do with combining shapes to make other shapes, much in the same way that we combined values of coins to make other values.

The most joyful moments during these days is having this opportunity to be a part of these early moments of learning about addition. When students say that they can’t get their numbers to add up to 100, though they know that they do, I can sit with them and help them sort out what’s going on. It’s so illuminating for me hear them tell me what they’ve done, and then to help them see another way of interacting with the numbers.

Addendum: as soon as this post went up the generous and brilliant connector-of-all -things-math offered me this link to some other coin projects

Simply awesome.


Re-Framing a Lesson

January 28, 2017

Bookmaking with First Graders

Bookmaking with First Graders

OMG Have I got a teaching tip for anyone who has ever pulled their hair out trying encourage students to make their drawings bigger, to fill up the page. It’s only taken me like 25 years of working with students to figure this out. This is big.

Filling up a page with a drawing

Filling up a page with a drawing

There’s this variations of a bookmaking project that I do with mostly first and second graders that includes a drawing. The bigger and bolder the drawing is, the better it looks in the book. Needless to say, it’s such a struggle for this age of student to make their drawings big enough.


Usually I give the students the paper that their drawing goes on and do everything but beg them to draw bigger. Well, sometimes I beg. Then, yesterday  (Friday)  Carter, a 7 year-old in my first class of the day, suggested that, before they start their drawing, I  lay the paper inside the frame that will surround it. It had never occurred to me to do this, so I tried it out in my next class of the day.


Unbelievable. In my next class, after sliding the paper behind the frame before the drawing began, every single student filled up the paper with large bold drawings to go along with their stories.

Never has this happened before.

Maybe it was just a fluke, maybe this class had been bribed enough times to fill up the page that they now did it instinctively. I had one more class to go.


Next class, same thing happened. They filled up the space with big drawings.


Some students lifted the frame away after the first part of the drawing was done so that they could make their drawings even bigger. OMG I was so happy. My conclusion: if you want students to make a drawing to fill up a space, FRAME THE SPACE with a dark frame! I don’t know why it works, but far be it from me to ever think I can fathom what goes on in the mind of a 7 year-old.

Now here’s the part that gives me chills…I have to ask myself, why did Carter put forth his suggestion? I give credit to this: recently I was impressed by reading Malke Rosenfeld’s book about engaging students in whole body learning. While I teach different subject matter than Malke, I am deeply impressed by how she gives her students permission to explore the learning space before she begins her lessons. I took this to heart, and this week, for the first time, within certain boundaries, I encouraged students to fold and unfold, then explore and examine the materials that we were using together. In some way I think this sense of engagement with the materials led Carter to making a suggestion that was based on what would have worked better for him. I already know that my best teaching tips come from the single digit crowd, I just don’t always know how to tap into them.

So thank you Malke, thank you Carter, and OMG I am so happy.


How to Make an Origami Toy Boat

How to Make an Origami Toy Boat

Full disclosure: I did not try out teaching 4-year olds how to make origami boats. It’s not that I didn’t want to, or that I chose not to, it was just that there were other things I wanted to do more, and my time with these little ones was limited. Much to my delight, though, after we used the boats in  our activity, the children asked me about how they were made. I did a demonstration, with hope that this may encourage an interest in  paper-folding.

Toy Boats

origami Boats, stacked

I chose to use these paper boats because they stack. Just for the record, I was curious to see if they floated. Turns out ;Yes! Until the paper absorbs too much water, these vessels are sea worthy. What was more useful for me, though, was that they can stand on their own, so that we could use them as playing pieces for a board game.

During my workshops with these children I noticed that even the most accomplished child in the group could not coordinate counting items with the movement of his hands. In other words, if there was a  pile of 8 stones, these children would end up counting inaccurately because their fingers would move out of sync with the numbers that they were reciting. I was really interested to see this, partially because  I’ve read that there is something about learning to play the piano that  helps children be better at math: this now makes sense to me, as playing notes would help train a person to coordinate fingers with intention.

Toy Boats and Board

Toy Boats and Board, drawn on large paper

Wanting to try out a simple, and, yes, frugal, made-from-paper activity to encourage accurate counting skills, I worked out a sweet game that the kids seemed to like . What we did mimics  classic board games where a die is thrown, and the player advances a certain number of spaces along a line. It was, however,  important to me that I didn’t want to create winners or losers.  This is how it went: the playing pieces were these paper boats, and when two boats land on the same space, they become a team, and stacked together. The point of the game is to get all the boats stacked together as a team before any boat reaches the end of the meandering number line, which, just, for no particular reason other than I ran out of space on my paper, was 42 units long.

Beginning at 0

Beginning at zero

Unfortunately, I didn’t get a photo of the kids playing this game, but, they played in groups of three and four, and they seemed to enjoy watching others play as well as playing the game themselves. Counting spaces, counting the dots on the dice, and (especially!) anticipating what throw of the die would yield the desired outcome were all challenging but doable for these kids.

Each of the five sessions that I worked with these students, one-third of my lesson plan was to focused my interpretation of relationship thinking, such as creating patterns from shaped paper,  developing finger sense, estimating, discussing what was the same and different about shapes and flowers, and this unit counting game.

Other parts of my time of my time with these kids was artful numbers, which I what my next and last post about my time with these students will be about.

I had to learn how to make these origami boats for this project. I looked many different models, but this one that I’ve shown  I found most enchanting. I put together a video of it, that is worth watching because there’s some pointers included that I just can’t fit onto a tutorial page.

Happy boating.

Addendum 5/6/2017 Just came across a tutorial that shows an alternative way of making this boat. An interesting way of altering the folding method by Gregor Müller

Finger Tracing

June 16, 2016

Hand of a four year old

Hand tracing by a four-year-old

This post is for people who have the good fortune to be working with four- and five-year olds.

For quite some time I’ve been exploring ways of drawing the attention of young students to their fingers. In my post Counting to Ten, March 2015  I wrote”My thinking here is that I want these students to create a visual that connects the numbers that they are learning to the fingers that they count on.” I see the fingers as the original number line.

Recently I watched a TedX  Stanford video of Jo Boaler, an educator who is involved in research in how to support learning and growth. I’m already of fan of Jo Boaler, but particularly liked this talk of hers and particular like what she says about counting on fingers. Starting at about the point 7:22 on the video, she tells the audience that when we calculate, the brain area that sees fingers lights up. She then goes on to tell us that the amount of finger perception grade 1 students have is a better predictor of math achievement in grade 2 than test scores.

Hand tracing with a cut-out tunnel to fit over the student's wrist.

Hand tracing with a cut-out tunnel to fit over the student’s wrist.

After hearing Jo Boaler’s talk, I couldn’t help but wonder if it is possible to modify my finger tracing project in a way that could possibly help children strengthen their finger perception? This isn’t something that I’ve ever thought about before, but I guess that some people are more challenged than others in connecting the sensation of having a finger touched to the finger that is being touched. More simply said, when I touch your finger, do you know, without peeking, which finger I am touching?

First we did some counting together, up to five. Then the children traced their hands. The adults wrote down the numbers on the drawn fingers. We cut out a tunnel so the child’s hand could rest under the drawing. Then a partner would touch a finger, and the child would reference the drawing and say which finger had been touched.

Here’s a few short video clips of how it went:

Mostly it was children doing this with each other, but it was easier for me to use these clips that show the adults working with the students. When the students were working with each other I put little stickers on the student’s hands, labeling the fingers 1-5 so that the student who was doing the finger touching would know what number was matched to the finger that he or she was touching.

It was interesting to see that there was a huge range between how hard and how easy it was for children to identify which finger was touched. One child simply could not make the connection at all. It finally occurred to me to let her see her hand and the map of her hand at the same time, to see if she could develop the connection. This seemed to work out for her. Here’s what it looked like towards the end:

This was my first attempt at this kind of…um….hands-on drawing project/ finger game with young students. It was a really quick, let’s-see-what-happens-if-we-do- this kind of thing, but there seems to be something interesting going on here. and I hope to do more of this, and hope other people will try it out too! Thanks Jo Boaler!

Addendum July 21, 2016

I’m seeing a trend… my addendums are getting more frequent and could become posts in themselves. Seems like once I post something I stumble across something that’s totally relevant, or someone tags me with content that applies. This addendum has both.

Part 1

First, here’s an absolutely accessible, research based activity to do with young children, to help their brains develop its mathematical regions. It’s based on the idea that there’s a correlation between preschooler’s sense of approximation and their general ability to do mathematics. It looks like this:

from a report on the work of Jinjing Wang of John Hopkins Universitys

from a report on the work of Jinjing Wang of John Hopkins University

Looks to me that there’s lots of possibilities for here for books that I could make with kids for them to  bring home. Even though the task here is to estimate which side of the page has more dots, I can see all sort of other kinds of vocabulary that can be come up with 4 year olds that this kind of graphic can facilitate. Here’s a link the the article about this, that Dave Radcliffe @daveinstpaul pointed out to the twitter community:–WJ4LCbVq8EZ

Part 2

Ted Lewis saw my post and alerted me to his, which, discussed the  longer and more satisfying exploration and examination of ” number sense and how we create it.” This is the graphic that accompanied Ted’s post:


 Okay, this is a perfect reminder that I don’t need a computer screen and Big Bird and Elmo to do this kind of work with students. Ted is using the inequality signs for this exercise. (Silly Ted, inequality signs are the downfall of many, but let’s talk about that some other time….)  The point though, is, if you are interested in visuals and the brain and the want some insights and food for thought, read this post for yourself .

I’ll be working, weekly, with four-year-olds starting the first week of July. Can’t wait to see if I can corral them into doing any of the activities that are inspired by these dots and other ways of thinking.


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